Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 262191 10 pageshttpdxdoiorg1011552013262191

Research ArticleApproximate Controllability of Sobolev Type NonlocalFractional Stochastic Dynamic Systems in Hilbert Spaces

Mourad Kerboua1 Amar Debbouche1 and Dumitru Baleanu234

1 Department of Mathematics Guelma University 24000 Guelma Algeria2 Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz UniversityPO Box 80204 Jeddah 21589 Saudi Arabia

3 Department of Mathematics and Computer Sciences Cankaya University 06530 Ankara Turkey4 Institute of Space Sciences Magurele Bucharest Romania

Correspondence should be addressed to Amar Debbouche amar debboucheyahoofr

Received 19 July 2013 Accepted 27 September 2013

Academic Editor Bashir Ahmad

Copyright copy 2013 Mourad Kerboua et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study a class of fractional stochastic dynamic control systems of Sobolev type in Hilbert spaces We use fixed point techniquefractional calculus stochastic analysis and methods adopted directly from deterministic control problems for the main results Anew set of sufficient conditions for approximate controllability is formulated and proved An example is also given to provide theobtained theory

1 Introduction

We are concerned with the following nonlocal fractionalstochastic system of Sobolev type

119862119863119902

119905[119871119909 (119905)] = 119872119909 (119905) + 119861119906 (119905) + 119891 (119905 119909 (119905))

+ 120590 (119905 119909 (119905))119889119908 (119905)

119889119905

119909 (0) + 119892 (119909 (119905)) = 1199090

(1)

where 119862119863119902119905is the Caputo fractional derivative of order 119902

0 lt 119902 le 1 and 119905 isin 119869 = [0 119887] Let 119883 and 119884 be two Hilbertspaces and the state 119909(sdot) takes its values in119883 We assume thatthe operators119871 and119872 are defined on domains contained in119883and ranges contained in 119884 the control function 119906(sdot) belongsto the space 1198712

Γ(119869 119880) a Hilbert space of admissible control

functions with119880 as a Hilbert space and 119861 is a bounded linearoperator from 119880 into 119884 It is also assumed that 119891 119869 times 119883 rarr119884 119892 119862(119869 119883) rarr 119884 and 120590 119869 times 119883 rarr 119871

0

2are appropriate

functions 1199090is Γ0measurable 119883-valued random variables

independent of119908 Here Γ Γ0 11987102 and119908will be specified later

The field of fractional differential equations and itsapplications has gained a lot of importance during the pastthree decades mainly because it has become a powerfultool in modeling several complex phenomena in numerousseemingly diverse and widespread fields of science andengineering [1ndash8] Recently there has been a significantdevelopment in the existence and uniqueness of solutions ofinitial and boundary value problem for fractional evolutionsystems [9]

Controllability is one of the important fundamental con-cepts in mathematical control theory and plays a vital role inboth deterministic and stochastic control systems Since thecontrollability notion has extensive industrial and biologicalapplications in the literature there aremany different notionsof controllability both for linear and nonlinear dynamicalsystems Controllability of the deterministic and stochasticdynamical control systems in infinite dimensional spaces iswell developed using different kinds of approaches It shouldbe mentioned that the theory of controllability for nonlinearfractional dynamical systems is still in the initial stage Thereare fewworks in controllability problems for different kinds ofsystems described by fractional differential equations [10 11]

2 Abstract and Applied Analysis

The exact controllability for semilinear fractional ordersystem when the nonlinear term is independent of thecontrol function is proved by many authors [12ndash15] In thesepapers the authors have proved the exact controllability byassuming that the controllability operator has an inducedinverse on a quotient space However if the semigroupassociatedwith the system is compact then the controllabilityoperator is also compact and hence the induced inverse doesnot exist because the state space is infinite dimensional [16]Thus the concept of exact controllability is too strong andhas limited applicability and the approximate controllabilityis a weaker concept than complete controllability and it iscompletely adequate in applications for these control systems

In [17 18] the approximate controllability of first orderdelay control systems has been proved when nonlinear termis a function of both state function and control function byassuming that the corresponding linear system is approxi-mately controllable To prove the approximate controllabilityof first order system with or without delay a relation betweenthe reachable set of a semilinear system and that of thecorresponding linear system is proved in [19ndash23] There areseveral papers devoted to the approximate controllabilityfor semilinear control systems when the nonlinear term isindependent of control function [24ndash27]

Stochastic differential equations have attracted greatinterest due to their applications in various fields of scienceand engineering There are many interesting results on thetheory and applications of stochastic differential equations(see [12 28ndash32] and the references therein) To build morerealistic models in economics social sciences chemistryfinance physics and other areas stochastic effects need tobe taken into account Therefore many real world problemscan be modeled by stochastic differential equations Thedeterministic models often fluctuate due to noise so wemust move from deterministic control to stochastic controlproblems

In the present literature there are only a limited numberof papers that deal with the approximate controllability offractional stochastic systems [33] as well as with the existenceand controllability results of fractional evolution equations ofSobolev type [34]

Sakthivel et al [35] studied the approximate control-lability of a class of dynamic control systems describedby nonlinear fractional stochastic differential equations inHilbert spaces More recent works can be found in [10 11]Debbouche et al [4] established a class of fractional nonlocalnonlinear integrodifferential equations of Sobolev type usingnew solution operators Feckan et al [36] presented the con-trollability results corresponding to two admissible controlsets for fractional functional evolution equations of Sobolevtype in Banach spaces with the help of two new characteristicsolution operators and their properties such as boundednessand compactness

It should be mentioned that there is no work yet reportedon the approximate controllability of Sobolev type fractionaldeterministic stochastic control systems Motivated by theabove facts in this paper we establish the approximate con-trollability for a class of fractional stochastic dynamic systemsof Sobolev Type with nonlocal conditions in Hilbert spaces

The paper is organized as follows in Section 2 we presentsome essential facts in fractional calculus semigroup theorystochastic analysis and control theory that will be used toobtain our main results In Section 3 we state and proveexistence and approximate controllability results for Sobolevtype fractional stochastic system (1) The last sections dealwith an illustrative example and a discussion for possiblefuture work in this direction

2 Preliminaries

In this section we give some basic definitions notationsproperties and lemmas which will be used throughout thework In particular we state main properties of fractionalcalculus [37ndash40] well-known facts in semigroup theory [41ndash43] and elementary principles of stochastic analysis [31 44]

Definition 1 The fractional integral of order 120572 gt 0 of afunction 119891 isin 1198711([119886 119887]R+) is given by

119868120572

119886119891 (119905) =

1

Γ (120572)int

119905

119886

(119905 minus 119904)120572minus1119891 (119904) 119889119904 (2)

where Γ is the gamma function If 119886 = 0 we canwrite 119868120572119891(119905) =(119892120572lowast 119891)(119905) where

119892120572 (119905) =

1

Γ (120572)119905120572minus1 119905 gt 0

0 119905 le 0

(3)

and as usual lowast denotes the convolution of functions More-over lim

120572rarr0119892120572(119905) = 120575(119905) with 120575 the delta Dirac function

Definition 2 The Riemann-Liouville derivative of order 119899 minus1 lt 120572 lt 119899 119899 isin N for a function 119891 isin 119862([0infin)) is given by

119871119863120572119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

119891 (119904)

(119905 minus 119904)120572+1minus119899

119889119904 119905 gt 0 (4)

Definition 3 The Caputo derivative of order 119899 minus 1 lt 120572 lt 119899119899 isin N for a function 119891 isin 119862([0infin)) is given by

119862119863120572119891 (119905) =

119871119863120572

(119891 (119905) minus

119899minus1

sum

119896=0

119905119896

119896119891(119896)(0)) 119905 gt 0 (5)

Remark 4 The following properties hold (see eg [45])

(i) If 119891 isin 119862119899([0infin)) then

119862119863120572119891 (119905) =

1

Γ (119899 minus 120572)int

119905

0

119891(119899)(119904)

(119905 minus 119904)120572+1minus119899

119889119904 = 119868119899minus120572119891119899(119905)

119905 gt 0 119899 minus 1 lt 120572 lt 119899 119899 isin N

(6)

(ii) The Caputo derivative of a constant is equal to zero(iii) If 119891 is an abstract function with values in119883 then the

integrals which appear in Definitions 1ndash3 are taken inBochnerrsquos sense

Abstract and Applied Analysis 3

We introduce the following assumptions on the operators119871 and119872

(H1) 119871 and119872 are linear operators and119872 is closed

(H2) 119863(119871) sub 119863(119872) and 119871 is bijective(H3) 119871minus1 119884 rarr 119863(119871) sub 119883 is a linear compact operator

Remark 5 From (H3) we deduce that 119871minus1 is a bounded

operator for short we denote 119862 = 119871minus1 Note (H

3) also

implies that 119871 is closed since 119871minus1 is closed and injectivethen its inverse is also closed It comes from (H

1)ndash(H3)

and the closed graph theorem we obtain the boundednessof the linear operator 119872119871minus1 119884 rarr 119884 Consequently119872119871minus1 generates a semigroup 119878(119905) = 119890119872119871

minus1

119905 119905 ge 0 We

suppose that 1198720 = sup119905ge0119878(119905) lt infin According to

previous definitions it is suitable to rewrite problem (1) asthe equivalent integral equation

119871119909 (119905) = 119871 [1199090 minus 119892 (119909)]

+1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [119872119909 (119904) + 119861119906 (119904) + 119891 (119904 119909 (119904))] 119889119904

+1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1120590 (119904 119909 (119904)) 119889119908 (119904)

(7)

provided the integral in (7) exists Before formulating thedefinition of mild solution of (1) we first give the followingdefinitions corollaries lemmas and notations

Let (Ω Γ 119875) be a complete probability space equippedwith a normal filtration Γ

119905 119905 isin 119869 satisfying the usual

conditions (ie right continuous and Γ0containing all 119875-null

sets) We consider four real separable spaces 119883 119884 119864 and 119880and 119876-Wiener process on (Ω Γ

119887 119875) with the linear bounded

covariance operator 119876 such that tr119876 lt infin We assume thatthere exists a complete orthonormal system 119890

119899119899ge1

in 119864 abounded sequence of nonnegative real numbers 120582

119899 such

that 119876119890119899= 120582119899119890119899 119899 = 1 2 and a sequence 120573

119899119899ge1

ofindependent Brownian motions such that

⟨119908 (119905) 119890⟩ =

infin

sum

119899=1

radic120582119899⟨119890119899 119890⟩ 120573119899 (119905) 119890 isin 119864 119905 isin 119869 (8)

and Γ119905= Γ119908

119905 where Γ119908

119905is the sigma algebra generated

by 119908(119904) 0 le 119904 le 119905 Let 11987102= 1198712(11987612119864119883) be

the space of all Hilbert-Schmidt operators from 11987612119864 to 119883with the inner product ⟨120595 120587⟩1198710

2= tr[120595119876120587lowast] Let 1198712(Γ

119887 119883)

be the Banach space of all Γ119887-measurable square integrablerandom variables with values in the Hilbert space119883 Let 119864(sdot)denote the expectation with respect to the measure 119875 Let119862(119869 119871

2(Γ 119883)) be theHilbert space of continuousmaps from 119869

into 1198712(Γ 119883) satisfying sup119905isin119869119864119909(119905)

2lt infin Let1198672(119869 119883) be

a closed subspace of 119862(119869 1198712(Γ 119883)) consisting of measurableand Γ119905-adapted119883-valued process 119909 isin 119862(119869 1198712(Γ 119883)) endowed

with the norm 1199091198672

= (sup119905isin119869119864119909(119905)

2

119883)12 For details we

refer the reader to [35 44] and references thereinThe following results will be used through out this paper

Lemma 6 (see [33]) Let 119866 119869 times Ω rarr 1198710

2be a strongly

measurable mapping such that int1198870119864119866(119905)

119901

1198710

2

119889119905 lt infin Then

119864

10038171003817100381710038171003817100381710038171003817

int

119905

0

119866(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

119901

le 119871119866int

119905

0

119864119866(119904)119901

1198710

2

119889119904 (9)

for all 0 le 119905 le 119887 and 119901 ge 2 where 119871119866 is the constant involving

119901 and 119887

Now we present the mild solution of the problem (1)

Definition 7 (compare with [46 47] and [36 45]) A stochas-tic process 119909 isin 1198672(119869 119883) is a mild solution of (1) if foreach control 119906 isin 1198712

Γ(119869 119880) it satisfies the following integral

equation

119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906 (119904) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(10)

where S(119905) andT(119905) are characteristic operators given by

S (119905) = intinfin

0

119871minus1120585119902 (120579) 119878 (119905

119902120579) 119889120579

T (119905) = 119902intinfin

0

119871minus1120579120585119902 (120579) 119878 (119905

119902120579) 119889120579

(11)

Here 119878(119905) is a 1198620-semigroup generated by the linearoperator119872119871minus1 119884 rarr 119884 120585

119902is a probability density function

defined on (0infin) that is 120585119902(120579) ge 0 120579 isin (0infin) and

intinfin

0120585119902(120579)119889120579 = 1

Lemma 8 (see [45 48 49]) The operators S(119905)119905ge0

andT(119905)

119905ge0are strongly continuous that is for 119909 isin 119883 and

0 le 1199051 lt 1199052 le 119887 one has S(1199052)119909 minus S(1199051)119909 rarr 0 andT(1199052)119909 minusT(1199051)119909 rarr 0 as 119905

2rarr 1199051

We impose the following conditions on data of theproblem

(i) For any fixed 119905 ge 0T(119905) andS(119905) are bounded linearoperators that is for any 119909 isin 119883

T (119905) 119909 le 1198621198720 119909 S (119905) 119909 le1198621198720

Γ (119902)119909 (12)

(ii) The functions 119891 119869 times 119883 rarr 119884 120590 119869 times 119883 rarr 1198710

2and

119892 119862(119869 119883) rarr 119884 satisfy linear growth and Lipschitz

4 Abstract and Applied Analysis

conditions Moreover there exist positive constants1198731 1198732 gt 0 1198711 1198712 gt 0 and 1198961 1198962 gt 0 such that

1003817100381710038171003817119891(119905 119909) minus 119891(119905 119910)1003817100381710038171003817

2le 1198731

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119891(119905 119909)1003817100381710038171003817

2le 1198732(1 + 119909

2)

1003817100381710038171003817120590(119905 119909) minus 120590(119905 119910)1003817100381710038171003817

2

1198710

2

le 1198711

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

120590(119905 119909)2

1198710

2

le 1198712(1 + 119909

2)

1003817100381710038171003817119892(119909) minus 119892(119910)1003817100381710038171003817

2le 1198961

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119892(119909)1003817100381710038171003817

2le 1198962 (1 + 119909

2)

(13)

(iii) The linear stochastic system is approximately control-lable on 119869

For each 0 le 119905 lt 119887 the operator 120572(120572119868 + Ψ1198870)minus1rarr 0 in

the strong operator topology as 120572 rarr 0+ where Ψ119887

0= int119887

0(119887 minus

119904)2(119902minus1)

T(119887minus119904)119861119861lowastTlowast(119887minus119904)119889119904 is the controllability GramianHere 119861lowast denotes the adjoint of 119861 andTlowast(119905) is the adjoint ofT(119905)

Observe that Sobolev type nonlocal linear fractionaldeterministic control system

119862119863119902

119905[119871119909 (119905)] = 119872119909 (119905) + 119861119906 (119905) 119905 isin 119869

119909 (0) + 119892 (119909 (119905)) = 1199090

(14)

corresponding to (1) is approximately controllable on 119869 iffthe operator 120572(120572119868 + Ψ119887

0)minus1rarr 0 strongly as 120572 rarr 0

+ Theapproximate controllability for linear fractional deterministiccontrol system (14) is a natural generalization of approximatecontrollability of linear first order control system (119902 = 1119892 = 0 and 119871 is the identity) [50]

Definition 9 The system (1) is approximately controllable on119869 ifR(119887) = 1198712(Ω Γ

119887 119883) where

R (119887) = 119909 (119887) = 119909 (119887 119906) 119906 isin 1198712

Γ(119869 119880) (15)

Here 1198712Γ(119869 119880) is the closed subspace of 1198712

Γ(119869 times Ω119880)

consisting of all Γ119905-adapted 119880-valued stochastic processes

The following lemma is required to define the controlfunction [35]

Lemma 10 For any 119909119887isin 1198712(Γ119887 119883) there exists 120593 isin

1198712

Γ(Ω 1198712(0 119887 119871

0

2)) such that 119909

119887= 119864119909119887+ int119887

0120593(119904)119889119908(119904)

Now for any 120572 gt 0 and 119909119887isin 1198712(Γ119887 119883) one defines the

control function in the following form

119906120572(119905 119909) = 119861

lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times [(120572119868 + Ψ119887

0)minus1

119864119909119887 minusS (119887) 119871 [1199090 minus 119892 (119909)]

+int

119905

0

(120572119868 + Ψ119887

0)minus1

120593 (119904) 119889119908 (119904)]

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909 (119904)) 119889119904

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

(16)

Lemma 11 There exist positive real constants such thatfor all 119909 119910 isin 119867

2 one has

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2le 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (17)

1198641003817100381710038171003817119906120572(119905 119909)

1003817100381710038171003817

2le (

1

119887+ 119864119909(119905)

2) (18)

Proof We start to prove (17) Let 119909 119910 isin 1198672 from Holderrsquos

inequality Lemma 6 and the assumption on the data weobtain

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2

le 3119864100381710038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905) (120572119868 + Ψ

119887

0)minus1

timesS (119887) 119871 [119892 (119909 (119905)) minus 119892 (119910 (119905))]100381710038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [119891 (119904 119909 (119904)) minus 119891 (119904 119910 (119904))] 11988911990410038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [120590 (119904 119909 (119904)) minus 120590 (119904 119910 (119904))] 119889119908 (119904)10038171003817100381710038171003817

2

Abstract and Applied Analysis 5

le3

12057221198612(119887)2119902minus2(1198621198720)

2(1198621198720

Γ(119902))

2

119871211989611198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198731

times int

119905

0

1198641003817100381710038171003817119909 (119904) minus 119910 (119904)

1003817100381710038171003817

2119889119904

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198711

times int

119905

0

1198641003817100381710038171003817119909(119904) minus 119910(119904)

1003817100381710038171003817

2119889119904

le 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(19)

where = (31205722)1198612(119887)2119902minus2(119862119872

0)2(1198621198720Γ(119902))

211987121198961+

(1198621198720Γ(119902))

4(1198872119902minus1(2119902 minus 1))119887[119873

1+ 1198711] The proof of the

inequality (18) can be established in a similar way to that of(17)

3 Approximate Controllability

In this section we formulate and prove conditions forthe existence and approximate controllability results of thenonlocal fractional stochastic dynamic control system ofSobolev type (1) using the contractionmapping principle Forany 120572 gt 0 define the operator 119865120572 1198672 rarr 1198672 by

119865120572119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906120572(119904 119909) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(20)

We state and prove the following lemma which will beused for the main results

Lemma 12 For any 119909 isin 1198672 119865120572(119909)(119905) is continuous on 119869 in

1198712-sense

Proof Let 0 le 1199051lt 1199052le 119887 Then for any fixed 119909 isin 119867

2 from

(20) we have

1198641003817100381710038171003817(119865120572119909)(1199052) minus (119865120572119909)(1199051)

1003817100381710038171003817

2le 4[

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(1199052) minus Π119909

119894(1199051)1003817100381710038171003817

2]

(21)

We begin with the first term and get

1198641003817100381710038171003817Π119909

1(1199052) minus Π

119909

1(1199051)1003817100381710038171003817

2= 1198641003817100381710038171003817(S(1199052) minus S(1199051))119871[1199090 minus 119892(119909)]

1003817100381710038171003817

2

le 1198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)]

times 1198641003817100381710038171003817S(1199052) minusS(1199051)

1003817100381710038171003817

2

(22)

The strong continuity of S(119905) implies that the right-handside of the last inequality tends to zero as 119905

2minus 1199051rarr 0

Next it follows fromHolderrsquos inequality and assumptionson the data that

1198641003817100381710038171003817Π119909

2(1199052) minus Π

119909

2(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052 minus 119904)119902minus1

T (1199052 minus 119904) 119861119906120572(119904 119909) 119889119904

minusint

1199051

0

(1199051 minus 119904)119902minus1

T(1199051 minus 119904)119861119906120572(119904 119909)119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051 minus 119904)119902minus1(T (1199052 minus 119904) minusT (1199051 minus 119904))

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)T (119905

2minus 119904)

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T(1199052minus 119904)119861119906

120572(119904 119909)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))119861119906

120572(119904 119909)119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

1198612(int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

1198612int

1199052

1199051

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904

(23)

Also we have

1198641003817100381710038171003817Π119909

3(1199052) minus Π119909

3(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

minusint

1199051

0

(1199051minus 119904)119902minus1

T(1199051minus 119904)119891(119904 119909(119904))119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

6 Abstract and Applied Analysis

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)

timesT (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891(119904 119909(119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T (1199052 minus 119904) minusT (1199051 minus 119904)) 119891 (119904 119909 (119904)) 119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

int

1199052

1199051

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904

(24)

Furthermore we use Lemma6 andprevious assumptionswe obtain

1198641003817100381710038171003817Π119909

4(1199052) minus Π119909

4(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

minusint

1199051

0

(1199051minus 119904)119902minus1

T (1199051minus 119904) 120590 (119904 119909 (119904)) 119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)T (1199052 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052 minus 119904)119902minus1

T(1199052 minus 119904)120590(119904 119909(119904))119889119908(119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 119871120590

1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))120590(119904 119909(119904))119889119904

1003817100381710038171003817

2

+ 119871120590(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817T (1199052 minus 119904) 120590 (119904 119909 (119904))

1003817100381710038171003817

2119889119904)

+ 119871120590

(1199052 minus 1199051)2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

times int

1199052

1199051

1198641003817100381710038171003817T(1199052 minus 119904)120590(119904 119909(119904))

1003817100381710038171003817

2119889119904

(25)

Hence using the strong continuity ofT(119905) and Lebesguersquosdominated convergence theorem we conclude that the right-hand side of the previous inequalities tends to zero as 119905

2minus

1199051rarr 0 Thus we conclude 119865

120572(119909)(119905) is continuous from

the right of [0 119887) A similar argument shows that it is alsocontinuous from the left of (0 119887]

Theorem 13 Assume hypotheses (i) and (ii) are satisfiedThenthe system (1) has a mild solution on 119869

Proof We prove the existence of a fixed point of the operator119865120572by using the contractionmapping principle First we show

that 119865120572(1198672) sub 119867

2 Let 119909 isin 119867

2 From (20) we obtain

1198641003817100381710038171003817119865120572119909(119905)

1003817100381710038171003817

2le 4[sup

119905isin119869

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2] (26)

Using assumptions (i)-(ii) Lemma 11 and standard com-putations yields

sup119905isin119869

1198641003817100381710038171003817Π119909

1(119905)1003817100381710038171003817

2le 11986221198722

01198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)] (27)

sup119905isin119869

4

sum

119894=2

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2

le (1198621198720

Γ(119902))

21198872119902minus1

2119902 minus 11198612 (

1

119887+ 1199092

1198672

)

+ (1198621198720

Γ (119902))

2

[1198872119902minus1

2119902 minus 11198732 minus

1198872119902minus1

2119902 minus 11198712119871120590] (1 + 119909

2

1198672

)

(28)

Hence (26)ndash(28) imply that 11986410038171003817100381710038171198651205721199091003817100381710038171003817

2

1198672

lt infin ByLemma 12 119865

120572119909 isin 119867

2 Thus for each 120572 gt 0 the operator

119865120572maps 119867

2into itself Next we use the Banach fixed point

theorem to prove that 119865120572has a unique fixed point in119867

2 We

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

The exact controllability for semilinear fractional ordersystem when the nonlinear term is independent of thecontrol function is proved by many authors [12ndash15] In thesepapers the authors have proved the exact controllability byassuming that the controllability operator has an inducedinverse on a quotient space However if the semigroupassociatedwith the system is compact then the controllabilityoperator is also compact and hence the induced inverse doesnot exist because the state space is infinite dimensional [16]Thus the concept of exact controllability is too strong andhas limited applicability and the approximate controllabilityis a weaker concept than complete controllability and it iscompletely adequate in applications for these control systems

In [17 18] the approximate controllability of first orderdelay control systems has been proved when nonlinear termis a function of both state function and control function byassuming that the corresponding linear system is approxi-mately controllable To prove the approximate controllabilityof first order system with or without delay a relation betweenthe reachable set of a semilinear system and that of thecorresponding linear system is proved in [19ndash23] There areseveral papers devoted to the approximate controllabilityfor semilinear control systems when the nonlinear term isindependent of control function [24ndash27]

Stochastic differential equations have attracted greatinterest due to their applications in various fields of scienceand engineering There are many interesting results on thetheory and applications of stochastic differential equations(see [12 28ndash32] and the references therein) To build morerealistic models in economics social sciences chemistryfinance physics and other areas stochastic effects need tobe taken into account Therefore many real world problemscan be modeled by stochastic differential equations Thedeterministic models often fluctuate due to noise so wemust move from deterministic control to stochastic controlproblems

In the present literature there are only a limited numberof papers that deal with the approximate controllability offractional stochastic systems [33] as well as with the existenceand controllability results of fractional evolution equations ofSobolev type [34]

Sakthivel et al [35] studied the approximate control-lability of a class of dynamic control systems describedby nonlinear fractional stochastic differential equations inHilbert spaces More recent works can be found in [10 11]Debbouche et al [4] established a class of fractional nonlocalnonlinear integrodifferential equations of Sobolev type usingnew solution operators Feckan et al [36] presented the con-trollability results corresponding to two admissible controlsets for fractional functional evolution equations of Sobolevtype in Banach spaces with the help of two new characteristicsolution operators and their properties such as boundednessand compactness

It should be mentioned that there is no work yet reportedon the approximate controllability of Sobolev type fractionaldeterministic stochastic control systems Motivated by theabove facts in this paper we establish the approximate con-trollability for a class of fractional stochastic dynamic systemsof Sobolev Type with nonlocal conditions in Hilbert spaces

The paper is organized as follows in Section 2 we presentsome essential facts in fractional calculus semigroup theorystochastic analysis and control theory that will be used toobtain our main results In Section 3 we state and proveexistence and approximate controllability results for Sobolevtype fractional stochastic system (1) The last sections dealwith an illustrative example and a discussion for possiblefuture work in this direction

2 Preliminaries

In this section we give some basic definitions notationsproperties and lemmas which will be used throughout thework In particular we state main properties of fractionalcalculus [37ndash40] well-known facts in semigroup theory [41ndash43] and elementary principles of stochastic analysis [31 44]

Definition 1 The fractional integral of order 120572 gt 0 of afunction 119891 isin 1198711([119886 119887]R+) is given by

119868120572

119886119891 (119905) =

1

Γ (120572)int

119905

119886

(119905 minus 119904)120572minus1119891 (119904) 119889119904 (2)

where Γ is the gamma function If 119886 = 0 we canwrite 119868120572119891(119905) =(119892120572lowast 119891)(119905) where

119892120572 (119905) =

1

Γ (120572)119905120572minus1 119905 gt 0

0 119905 le 0

(3)

and as usual lowast denotes the convolution of functions More-over lim

120572rarr0119892120572(119905) = 120575(119905) with 120575 the delta Dirac function

Definition 2 The Riemann-Liouville derivative of order 119899 minus1 lt 120572 lt 119899 119899 isin N for a function 119891 isin 119862([0infin)) is given by

119871119863120572119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

119891 (119904)

(119905 minus 119904)120572+1minus119899

119889119904 119905 gt 0 (4)

Definition 3 The Caputo derivative of order 119899 minus 1 lt 120572 lt 119899119899 isin N for a function 119891 isin 119862([0infin)) is given by

119862119863120572119891 (119905) =

119871119863120572

(119891 (119905) minus

119899minus1

sum

119896=0

119905119896

119896119891(119896)(0)) 119905 gt 0 (5)

Remark 4 The following properties hold (see eg [45])

(i) If 119891 isin 119862119899([0infin)) then

119862119863120572119891 (119905) =

1

Γ (119899 minus 120572)int

119905

0

119891(119899)(119904)

(119905 minus 119904)120572+1minus119899

119889119904 = 119868119899minus120572119891119899(119905)

119905 gt 0 119899 minus 1 lt 120572 lt 119899 119899 isin N

(6)

(ii) The Caputo derivative of a constant is equal to zero(iii) If 119891 is an abstract function with values in119883 then the

integrals which appear in Definitions 1ndash3 are taken inBochnerrsquos sense

Abstract and Applied Analysis 3

We introduce the following assumptions on the operators119871 and119872

(H1) 119871 and119872 are linear operators and119872 is closed

(H2) 119863(119871) sub 119863(119872) and 119871 is bijective(H3) 119871minus1 119884 rarr 119863(119871) sub 119883 is a linear compact operator

Remark 5 From (H3) we deduce that 119871minus1 is a bounded

operator for short we denote 119862 = 119871minus1 Note (H

3) also

implies that 119871 is closed since 119871minus1 is closed and injectivethen its inverse is also closed It comes from (H

1)ndash(H3)

and the closed graph theorem we obtain the boundednessof the linear operator 119872119871minus1 119884 rarr 119884 Consequently119872119871minus1 generates a semigroup 119878(119905) = 119890119872119871

minus1

119905 119905 ge 0 We

suppose that 1198720 = sup119905ge0119878(119905) lt infin According to

previous definitions it is suitable to rewrite problem (1) asthe equivalent integral equation

119871119909 (119905) = 119871 [1199090 minus 119892 (119909)]

+1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [119872119909 (119904) + 119861119906 (119904) + 119891 (119904 119909 (119904))] 119889119904

+1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1120590 (119904 119909 (119904)) 119889119908 (119904)

(7)

provided the integral in (7) exists Before formulating thedefinition of mild solution of (1) we first give the followingdefinitions corollaries lemmas and notations

Let (Ω Γ 119875) be a complete probability space equippedwith a normal filtration Γ

119905 119905 isin 119869 satisfying the usual

conditions (ie right continuous and Γ0containing all 119875-null

sets) We consider four real separable spaces 119883 119884 119864 and 119880and 119876-Wiener process on (Ω Γ

119887 119875) with the linear bounded

covariance operator 119876 such that tr119876 lt infin We assume thatthere exists a complete orthonormal system 119890

119899119899ge1

in 119864 abounded sequence of nonnegative real numbers 120582

119899 such

that 119876119890119899= 120582119899119890119899 119899 = 1 2 and a sequence 120573

119899119899ge1

ofindependent Brownian motions such that

⟨119908 (119905) 119890⟩ =

infin

sum

119899=1

radic120582119899⟨119890119899 119890⟩ 120573119899 (119905) 119890 isin 119864 119905 isin 119869 (8)

and Γ119905= Γ119908

119905 where Γ119908

119905is the sigma algebra generated

by 119908(119904) 0 le 119904 le 119905 Let 11987102= 1198712(11987612119864119883) be

the space of all Hilbert-Schmidt operators from 11987612119864 to 119883with the inner product ⟨120595 120587⟩1198710

2= tr[120595119876120587lowast] Let 1198712(Γ

119887 119883)

be the Banach space of all Γ119887-measurable square integrablerandom variables with values in the Hilbert space119883 Let 119864(sdot)denote the expectation with respect to the measure 119875 Let119862(119869 119871

2(Γ 119883)) be theHilbert space of continuousmaps from 119869

into 1198712(Γ 119883) satisfying sup119905isin119869119864119909(119905)

2lt infin Let1198672(119869 119883) be

a closed subspace of 119862(119869 1198712(Γ 119883)) consisting of measurableand Γ119905-adapted119883-valued process 119909 isin 119862(119869 1198712(Γ 119883)) endowed

with the norm 1199091198672

= (sup119905isin119869119864119909(119905)

2

119883)12 For details we

refer the reader to [35 44] and references thereinThe following results will be used through out this paper

Lemma 6 (see [33]) Let 119866 119869 times Ω rarr 1198710

2be a strongly

measurable mapping such that int1198870119864119866(119905)

119901

1198710

2

119889119905 lt infin Then

119864

10038171003817100381710038171003817100381710038171003817

int

119905

0

119866(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

119901

le 119871119866int

119905

0

119864119866(119904)119901

1198710

2

119889119904 (9)

for all 0 le 119905 le 119887 and 119901 ge 2 where 119871119866 is the constant involving

119901 and 119887

Now we present the mild solution of the problem (1)

Definition 7 (compare with [46 47] and [36 45]) A stochas-tic process 119909 isin 1198672(119869 119883) is a mild solution of (1) if foreach control 119906 isin 1198712

Γ(119869 119880) it satisfies the following integral

equation

119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906 (119904) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(10)

where S(119905) andT(119905) are characteristic operators given by

S (119905) = intinfin

0

119871minus1120585119902 (120579) 119878 (119905

119902120579) 119889120579

T (119905) = 119902intinfin

0

119871minus1120579120585119902 (120579) 119878 (119905

119902120579) 119889120579

(11)

Here 119878(119905) is a 1198620-semigroup generated by the linearoperator119872119871minus1 119884 rarr 119884 120585

119902is a probability density function

defined on (0infin) that is 120585119902(120579) ge 0 120579 isin (0infin) and

intinfin

0120585119902(120579)119889120579 = 1

Lemma 8 (see [45 48 49]) The operators S(119905)119905ge0

andT(119905)

119905ge0are strongly continuous that is for 119909 isin 119883 and

0 le 1199051 lt 1199052 le 119887 one has S(1199052)119909 minus S(1199051)119909 rarr 0 andT(1199052)119909 minusT(1199051)119909 rarr 0 as 119905

2rarr 1199051

We impose the following conditions on data of theproblem

(i) For any fixed 119905 ge 0T(119905) andS(119905) are bounded linearoperators that is for any 119909 isin 119883

T (119905) 119909 le 1198621198720 119909 S (119905) 119909 le1198621198720

Γ (119902)119909 (12)

(ii) The functions 119891 119869 times 119883 rarr 119884 120590 119869 times 119883 rarr 1198710

2and

119892 119862(119869 119883) rarr 119884 satisfy linear growth and Lipschitz

4 Abstract and Applied Analysis

conditions Moreover there exist positive constants1198731 1198732 gt 0 1198711 1198712 gt 0 and 1198961 1198962 gt 0 such that

1003817100381710038171003817119891(119905 119909) minus 119891(119905 119910)1003817100381710038171003817

2le 1198731

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119891(119905 119909)1003817100381710038171003817

2le 1198732(1 + 119909

2)

1003817100381710038171003817120590(119905 119909) minus 120590(119905 119910)1003817100381710038171003817

2

1198710

2

le 1198711

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

120590(119905 119909)2

1198710

2

le 1198712(1 + 119909

2)

1003817100381710038171003817119892(119909) minus 119892(119910)1003817100381710038171003817

2le 1198961

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119892(119909)1003817100381710038171003817

2le 1198962 (1 + 119909

2)

(13)

(iii) The linear stochastic system is approximately control-lable on 119869

For each 0 le 119905 lt 119887 the operator 120572(120572119868 + Ψ1198870)minus1rarr 0 in

the strong operator topology as 120572 rarr 0+ where Ψ119887

0= int119887

0(119887 minus

119904)2(119902minus1)

T(119887minus119904)119861119861lowastTlowast(119887minus119904)119889119904 is the controllability GramianHere 119861lowast denotes the adjoint of 119861 andTlowast(119905) is the adjoint ofT(119905)

Observe that Sobolev type nonlocal linear fractionaldeterministic control system

119862119863119902

119905[119871119909 (119905)] = 119872119909 (119905) + 119861119906 (119905) 119905 isin 119869

119909 (0) + 119892 (119909 (119905)) = 1199090

(14)

corresponding to (1) is approximately controllable on 119869 iffthe operator 120572(120572119868 + Ψ119887

0)minus1rarr 0 strongly as 120572 rarr 0

+ Theapproximate controllability for linear fractional deterministiccontrol system (14) is a natural generalization of approximatecontrollability of linear first order control system (119902 = 1119892 = 0 and 119871 is the identity) [50]

Definition 9 The system (1) is approximately controllable on119869 ifR(119887) = 1198712(Ω Γ

119887 119883) where

R (119887) = 119909 (119887) = 119909 (119887 119906) 119906 isin 1198712

Γ(119869 119880) (15)

Here 1198712Γ(119869 119880) is the closed subspace of 1198712

Γ(119869 times Ω119880)

consisting of all Γ119905-adapted 119880-valued stochastic processes

The following lemma is required to define the controlfunction [35]

Lemma 10 For any 119909119887isin 1198712(Γ119887 119883) there exists 120593 isin

1198712

Γ(Ω 1198712(0 119887 119871

0

2)) such that 119909

119887= 119864119909119887+ int119887

0120593(119904)119889119908(119904)

Now for any 120572 gt 0 and 119909119887isin 1198712(Γ119887 119883) one defines the

control function in the following form

119906120572(119905 119909) = 119861

lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times [(120572119868 + Ψ119887

0)minus1

119864119909119887 minusS (119887) 119871 [1199090 minus 119892 (119909)]

+int

119905

0

(120572119868 + Ψ119887

0)minus1

120593 (119904) 119889119908 (119904)]

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909 (119904)) 119889119904

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

(16)

Lemma 11 There exist positive real constants such thatfor all 119909 119910 isin 119867

2 one has

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2le 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (17)

1198641003817100381710038171003817119906120572(119905 119909)

1003817100381710038171003817

2le (

1

119887+ 119864119909(119905)

2) (18)

Proof We start to prove (17) Let 119909 119910 isin 1198672 from Holderrsquos

inequality Lemma 6 and the assumption on the data weobtain

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2

le 3119864100381710038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905) (120572119868 + Ψ

119887

0)minus1

timesS (119887) 119871 [119892 (119909 (119905)) minus 119892 (119910 (119905))]100381710038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [119891 (119904 119909 (119904)) minus 119891 (119904 119910 (119904))] 11988911990410038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [120590 (119904 119909 (119904)) minus 120590 (119904 119910 (119904))] 119889119908 (119904)10038171003817100381710038171003817

2

Abstract and Applied Analysis 5

le3

12057221198612(119887)2119902minus2(1198621198720)

2(1198621198720

Γ(119902))

2

119871211989611198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198731

times int

119905

0

1198641003817100381710038171003817119909 (119904) minus 119910 (119904)

1003817100381710038171003817

2119889119904

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198711

times int

119905

0

1198641003817100381710038171003817119909(119904) minus 119910(119904)

1003817100381710038171003817

2119889119904

le 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(19)

where = (31205722)1198612(119887)2119902minus2(119862119872

0)2(1198621198720Γ(119902))

211987121198961+

(1198621198720Γ(119902))

4(1198872119902minus1(2119902 minus 1))119887[119873

1+ 1198711] The proof of the

inequality (18) can be established in a similar way to that of(17)

3 Approximate Controllability

In this section we formulate and prove conditions forthe existence and approximate controllability results of thenonlocal fractional stochastic dynamic control system ofSobolev type (1) using the contractionmapping principle Forany 120572 gt 0 define the operator 119865120572 1198672 rarr 1198672 by

119865120572119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906120572(119904 119909) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(20)

We state and prove the following lemma which will beused for the main results

Lemma 12 For any 119909 isin 1198672 119865120572(119909)(119905) is continuous on 119869 in

1198712-sense

Proof Let 0 le 1199051lt 1199052le 119887 Then for any fixed 119909 isin 119867

2 from

(20) we have

1198641003817100381710038171003817(119865120572119909)(1199052) minus (119865120572119909)(1199051)

1003817100381710038171003817

2le 4[

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(1199052) minus Π119909

119894(1199051)1003817100381710038171003817

2]

(21)

We begin with the first term and get

1198641003817100381710038171003817Π119909

1(1199052) minus Π

119909

1(1199051)1003817100381710038171003817

2= 1198641003817100381710038171003817(S(1199052) minus S(1199051))119871[1199090 minus 119892(119909)]

1003817100381710038171003817

2

le 1198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)]

times 1198641003817100381710038171003817S(1199052) minusS(1199051)

1003817100381710038171003817

2

(22)

The strong continuity of S(119905) implies that the right-handside of the last inequality tends to zero as 119905

2minus 1199051rarr 0

Next it follows fromHolderrsquos inequality and assumptionson the data that

1198641003817100381710038171003817Π119909

2(1199052) minus Π

119909

2(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052 minus 119904)119902minus1

T (1199052 minus 119904) 119861119906120572(119904 119909) 119889119904

minusint

1199051

0

(1199051 minus 119904)119902minus1

T(1199051 minus 119904)119861119906120572(119904 119909)119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051 minus 119904)119902minus1(T (1199052 minus 119904) minusT (1199051 minus 119904))

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)T (119905

2minus 119904)

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T(1199052minus 119904)119861119906

120572(119904 119909)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))119861119906

120572(119904 119909)119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

1198612(int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

1198612int

1199052

1199051

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904

(23)

Also we have

1198641003817100381710038171003817Π119909

3(1199052) minus Π119909

3(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

minusint

1199051

0

(1199051minus 119904)119902minus1

T(1199051minus 119904)119891(119904 119909(119904))119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

6 Abstract and Applied Analysis

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)

timesT (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891(119904 119909(119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T (1199052 minus 119904) minusT (1199051 minus 119904)) 119891 (119904 119909 (119904)) 119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

int

1199052

1199051

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904

(24)

Furthermore we use Lemma6 andprevious assumptionswe obtain

1198641003817100381710038171003817Π119909

4(1199052) minus Π119909

4(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

minusint

1199051

0

(1199051minus 119904)119902minus1

T (1199051minus 119904) 120590 (119904 119909 (119904)) 119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)T (1199052 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052 minus 119904)119902minus1

T(1199052 minus 119904)120590(119904 119909(119904))119889119908(119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 119871120590

1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))120590(119904 119909(119904))119889119904

1003817100381710038171003817

2

+ 119871120590(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817T (1199052 minus 119904) 120590 (119904 119909 (119904))

1003817100381710038171003817

2119889119904)

+ 119871120590

(1199052 minus 1199051)2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

times int

1199052

1199051

1198641003817100381710038171003817T(1199052 minus 119904)120590(119904 119909(119904))

1003817100381710038171003817

2119889119904

(25)

Hence using the strong continuity ofT(119905) and Lebesguersquosdominated convergence theorem we conclude that the right-hand side of the previous inequalities tends to zero as 119905

2minus

1199051rarr 0 Thus we conclude 119865

120572(119909)(119905) is continuous from

the right of [0 119887) A similar argument shows that it is alsocontinuous from the left of (0 119887]

Theorem 13 Assume hypotheses (i) and (ii) are satisfiedThenthe system (1) has a mild solution on 119869

Proof We prove the existence of a fixed point of the operator119865120572by using the contractionmapping principle First we show

that 119865120572(1198672) sub 119867

2 Let 119909 isin 119867

2 From (20) we obtain

1198641003817100381710038171003817119865120572119909(119905)

1003817100381710038171003817

2le 4[sup

119905isin119869

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2] (26)

Using assumptions (i)-(ii) Lemma 11 and standard com-putations yields

sup119905isin119869

1198641003817100381710038171003817Π119909

1(119905)1003817100381710038171003817

2le 11986221198722

01198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)] (27)

sup119905isin119869

4

sum

119894=2

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2

le (1198621198720

Γ(119902))

21198872119902minus1

2119902 minus 11198612 (

1

119887+ 1199092

1198672

)

+ (1198621198720

Γ (119902))

2

[1198872119902minus1

2119902 minus 11198732 minus

1198872119902minus1

2119902 minus 11198712119871120590] (1 + 119909

2

1198672

)

(28)

Hence (26)ndash(28) imply that 11986410038171003817100381710038171198651205721199091003817100381710038171003817

2

1198672

lt infin ByLemma 12 119865

120572119909 isin 119867

2 Thus for each 120572 gt 0 the operator

119865120572maps 119867

2into itself Next we use the Banach fixed point

theorem to prove that 119865120572has a unique fixed point in119867

2 We

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

We introduce the following assumptions on the operators119871 and119872

(H1) 119871 and119872 are linear operators and119872 is closed

(H2) 119863(119871) sub 119863(119872) and 119871 is bijective(H3) 119871minus1 119884 rarr 119863(119871) sub 119883 is a linear compact operator

Remark 5 From (H3) we deduce that 119871minus1 is a bounded

operator for short we denote 119862 = 119871minus1 Note (H

3) also

implies that 119871 is closed since 119871minus1 is closed and injectivethen its inverse is also closed It comes from (H

1)ndash(H3)

and the closed graph theorem we obtain the boundednessof the linear operator 119872119871minus1 119884 rarr 119884 Consequently119872119871minus1 generates a semigroup 119878(119905) = 119890119872119871

minus1

119905 119905 ge 0 We

suppose that 1198720 = sup119905ge0119878(119905) lt infin According to

previous definitions it is suitable to rewrite problem (1) asthe equivalent integral equation

119871119909 (119905) = 119871 [1199090 minus 119892 (119909)]

+1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1

times [119872119909 (119904) + 119861119906 (119904) + 119891 (119904 119909 (119904))] 119889119904

+1

Γ (119902)int

119905

0

(119905 minus 119904)119902minus1120590 (119904 119909 (119904)) 119889119908 (119904)

(7)

provided the integral in (7) exists Before formulating thedefinition of mild solution of (1) we first give the followingdefinitions corollaries lemmas and notations

Let (Ω Γ 119875) be a complete probability space equippedwith a normal filtration Γ

119905 119905 isin 119869 satisfying the usual

conditions (ie right continuous and Γ0containing all 119875-null

sets) We consider four real separable spaces 119883 119884 119864 and 119880and 119876-Wiener process on (Ω Γ

119887 119875) with the linear bounded

covariance operator 119876 such that tr119876 lt infin We assume thatthere exists a complete orthonormal system 119890

119899119899ge1

in 119864 abounded sequence of nonnegative real numbers 120582

119899 such

that 119876119890119899= 120582119899119890119899 119899 = 1 2 and a sequence 120573

119899119899ge1

ofindependent Brownian motions such that

⟨119908 (119905) 119890⟩ =

infin

sum

119899=1

radic120582119899⟨119890119899 119890⟩ 120573119899 (119905) 119890 isin 119864 119905 isin 119869 (8)

and Γ119905= Γ119908

119905 where Γ119908

119905is the sigma algebra generated

by 119908(119904) 0 le 119904 le 119905 Let 11987102= 1198712(11987612119864119883) be

the space of all Hilbert-Schmidt operators from 11987612119864 to 119883with the inner product ⟨120595 120587⟩1198710

2= tr[120595119876120587lowast] Let 1198712(Γ

119887 119883)

be the Banach space of all Γ119887-measurable square integrablerandom variables with values in the Hilbert space119883 Let 119864(sdot)denote the expectation with respect to the measure 119875 Let119862(119869 119871

2(Γ 119883)) be theHilbert space of continuousmaps from 119869

into 1198712(Γ 119883) satisfying sup119905isin119869119864119909(119905)

2lt infin Let1198672(119869 119883) be

a closed subspace of 119862(119869 1198712(Γ 119883)) consisting of measurableand Γ119905-adapted119883-valued process 119909 isin 119862(119869 1198712(Γ 119883)) endowed

with the norm 1199091198672

= (sup119905isin119869119864119909(119905)

2

119883)12 For details we

refer the reader to [35 44] and references thereinThe following results will be used through out this paper

Lemma 6 (see [33]) Let 119866 119869 times Ω rarr 1198710

2be a strongly

measurable mapping such that int1198870119864119866(119905)

119901

1198710

2

119889119905 lt infin Then

119864

10038171003817100381710038171003817100381710038171003817

int

119905

0

119866(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

119901

le 119871119866int

119905

0

119864119866(119904)119901

1198710

2

119889119904 (9)

for all 0 le 119905 le 119887 and 119901 ge 2 where 119871119866 is the constant involving

119901 and 119887

Now we present the mild solution of the problem (1)

Definition 7 (compare with [46 47] and [36 45]) A stochas-tic process 119909 isin 1198672(119869 119883) is a mild solution of (1) if foreach control 119906 isin 1198712

Γ(119869 119880) it satisfies the following integral

equation

119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906 (119904) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(10)

where S(119905) andT(119905) are characteristic operators given by

S (119905) = intinfin

0

119871minus1120585119902 (120579) 119878 (119905

119902120579) 119889120579

T (119905) = 119902intinfin

0

119871minus1120579120585119902 (120579) 119878 (119905

119902120579) 119889120579

(11)

Here 119878(119905) is a 1198620-semigroup generated by the linearoperator119872119871minus1 119884 rarr 119884 120585

119902is a probability density function

defined on (0infin) that is 120585119902(120579) ge 0 120579 isin (0infin) and

intinfin

0120585119902(120579)119889120579 = 1

Lemma 8 (see [45 48 49]) The operators S(119905)119905ge0

andT(119905)

119905ge0are strongly continuous that is for 119909 isin 119883 and

0 le 1199051 lt 1199052 le 119887 one has S(1199052)119909 minus S(1199051)119909 rarr 0 andT(1199052)119909 minusT(1199051)119909 rarr 0 as 119905

2rarr 1199051

We impose the following conditions on data of theproblem

(i) For any fixed 119905 ge 0T(119905) andS(119905) are bounded linearoperators that is for any 119909 isin 119883

T (119905) 119909 le 1198621198720 119909 S (119905) 119909 le1198621198720

Γ (119902)119909 (12)

(ii) The functions 119891 119869 times 119883 rarr 119884 120590 119869 times 119883 rarr 1198710

2and

119892 119862(119869 119883) rarr 119884 satisfy linear growth and Lipschitz

4 Abstract and Applied Analysis

conditions Moreover there exist positive constants1198731 1198732 gt 0 1198711 1198712 gt 0 and 1198961 1198962 gt 0 such that

1003817100381710038171003817119891(119905 119909) minus 119891(119905 119910)1003817100381710038171003817

2le 1198731

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119891(119905 119909)1003817100381710038171003817

2le 1198732(1 + 119909

2)

1003817100381710038171003817120590(119905 119909) minus 120590(119905 119910)1003817100381710038171003817

2

1198710

2

le 1198711

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

120590(119905 119909)2

1198710

2

le 1198712(1 + 119909

2)

1003817100381710038171003817119892(119909) minus 119892(119910)1003817100381710038171003817

2le 1198961

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119892(119909)1003817100381710038171003817

2le 1198962 (1 + 119909

2)

(13)

(iii) The linear stochastic system is approximately control-lable on 119869

For each 0 le 119905 lt 119887 the operator 120572(120572119868 + Ψ1198870)minus1rarr 0 in

the strong operator topology as 120572 rarr 0+ where Ψ119887

0= int119887

0(119887 minus

119904)2(119902minus1)

T(119887minus119904)119861119861lowastTlowast(119887minus119904)119889119904 is the controllability GramianHere 119861lowast denotes the adjoint of 119861 andTlowast(119905) is the adjoint ofT(119905)

Observe that Sobolev type nonlocal linear fractionaldeterministic control system

119862119863119902

119905[119871119909 (119905)] = 119872119909 (119905) + 119861119906 (119905) 119905 isin 119869

119909 (0) + 119892 (119909 (119905)) = 1199090

(14)

corresponding to (1) is approximately controllable on 119869 iffthe operator 120572(120572119868 + Ψ119887

0)minus1rarr 0 strongly as 120572 rarr 0

+ Theapproximate controllability for linear fractional deterministiccontrol system (14) is a natural generalization of approximatecontrollability of linear first order control system (119902 = 1119892 = 0 and 119871 is the identity) [50]

Definition 9 The system (1) is approximately controllable on119869 ifR(119887) = 1198712(Ω Γ

119887 119883) where

R (119887) = 119909 (119887) = 119909 (119887 119906) 119906 isin 1198712

Γ(119869 119880) (15)

Here 1198712Γ(119869 119880) is the closed subspace of 1198712

Γ(119869 times Ω119880)

consisting of all Γ119905-adapted 119880-valued stochastic processes

The following lemma is required to define the controlfunction [35]

Lemma 10 For any 119909119887isin 1198712(Γ119887 119883) there exists 120593 isin

1198712

Γ(Ω 1198712(0 119887 119871

0

2)) such that 119909

119887= 119864119909119887+ int119887

0120593(119904)119889119908(119904)

Now for any 120572 gt 0 and 119909119887isin 1198712(Γ119887 119883) one defines the

control function in the following form

119906120572(119905 119909) = 119861

lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times [(120572119868 + Ψ119887

0)minus1

119864119909119887 minusS (119887) 119871 [1199090 minus 119892 (119909)]

+int

119905

0

(120572119868 + Ψ119887

0)minus1

120593 (119904) 119889119908 (119904)]

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909 (119904)) 119889119904

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

(16)

Lemma 11 There exist positive real constants such thatfor all 119909 119910 isin 119867

2 one has

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2le 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (17)

1198641003817100381710038171003817119906120572(119905 119909)

1003817100381710038171003817

2le (

1

119887+ 119864119909(119905)

2) (18)

Proof We start to prove (17) Let 119909 119910 isin 1198672 from Holderrsquos

inequality Lemma 6 and the assumption on the data weobtain

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2

le 3119864100381710038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905) (120572119868 + Ψ

119887

0)minus1

timesS (119887) 119871 [119892 (119909 (119905)) minus 119892 (119910 (119905))]100381710038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [119891 (119904 119909 (119904)) minus 119891 (119904 119910 (119904))] 11988911990410038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [120590 (119904 119909 (119904)) minus 120590 (119904 119910 (119904))] 119889119908 (119904)10038171003817100381710038171003817

2

Abstract and Applied Analysis 5

le3

12057221198612(119887)2119902minus2(1198621198720)

2(1198621198720

Γ(119902))

2

119871211989611198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198731

times int

119905

0

1198641003817100381710038171003817119909 (119904) minus 119910 (119904)

1003817100381710038171003817

2119889119904

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198711

times int

119905

0

1198641003817100381710038171003817119909(119904) minus 119910(119904)

1003817100381710038171003817

2119889119904

le 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(19)

where = (31205722)1198612(119887)2119902minus2(119862119872

0)2(1198621198720Γ(119902))

211987121198961+

(1198621198720Γ(119902))

4(1198872119902minus1(2119902 minus 1))119887[119873

1+ 1198711] The proof of the

inequality (18) can be established in a similar way to that of(17)

3 Approximate Controllability

In this section we formulate and prove conditions forthe existence and approximate controllability results of thenonlocal fractional stochastic dynamic control system ofSobolev type (1) using the contractionmapping principle Forany 120572 gt 0 define the operator 119865120572 1198672 rarr 1198672 by

119865120572119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906120572(119904 119909) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(20)

We state and prove the following lemma which will beused for the main results

Lemma 12 For any 119909 isin 1198672 119865120572(119909)(119905) is continuous on 119869 in

1198712-sense

Proof Let 0 le 1199051lt 1199052le 119887 Then for any fixed 119909 isin 119867

2 from

(20) we have

1198641003817100381710038171003817(119865120572119909)(1199052) minus (119865120572119909)(1199051)

1003817100381710038171003817

2le 4[

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(1199052) minus Π119909

119894(1199051)1003817100381710038171003817

2]

(21)

We begin with the first term and get

1198641003817100381710038171003817Π119909

1(1199052) minus Π

119909

1(1199051)1003817100381710038171003817

2= 1198641003817100381710038171003817(S(1199052) minus S(1199051))119871[1199090 minus 119892(119909)]

1003817100381710038171003817

2

le 1198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)]

times 1198641003817100381710038171003817S(1199052) minusS(1199051)

1003817100381710038171003817

2

(22)

The strong continuity of S(119905) implies that the right-handside of the last inequality tends to zero as 119905

2minus 1199051rarr 0

Next it follows fromHolderrsquos inequality and assumptionson the data that

1198641003817100381710038171003817Π119909

2(1199052) minus Π

119909

2(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052 minus 119904)119902minus1

T (1199052 minus 119904) 119861119906120572(119904 119909) 119889119904

minusint

1199051

0

(1199051 minus 119904)119902minus1

T(1199051 minus 119904)119861119906120572(119904 119909)119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051 minus 119904)119902minus1(T (1199052 minus 119904) minusT (1199051 minus 119904))

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)T (119905

2minus 119904)

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T(1199052minus 119904)119861119906

120572(119904 119909)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))119861119906

120572(119904 119909)119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

1198612(int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

1198612int

1199052

1199051

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904

(23)

Also we have

1198641003817100381710038171003817Π119909

3(1199052) minus Π119909

3(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

minusint

1199051

0

(1199051minus 119904)119902minus1

T(1199051minus 119904)119891(119904 119909(119904))119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

6 Abstract and Applied Analysis

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)

timesT (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891(119904 119909(119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T (1199052 minus 119904) minusT (1199051 minus 119904)) 119891 (119904 119909 (119904)) 119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

int

1199052

1199051

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904

(24)

Furthermore we use Lemma6 andprevious assumptionswe obtain

1198641003817100381710038171003817Π119909

4(1199052) minus Π119909

4(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

minusint

1199051

0

(1199051minus 119904)119902minus1

T (1199051minus 119904) 120590 (119904 119909 (119904)) 119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)T (1199052 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052 minus 119904)119902minus1

T(1199052 minus 119904)120590(119904 119909(119904))119889119908(119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 119871120590

1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))120590(119904 119909(119904))119889119904

1003817100381710038171003817

2

+ 119871120590(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817T (1199052 minus 119904) 120590 (119904 119909 (119904))

1003817100381710038171003817

2119889119904)

+ 119871120590

(1199052 minus 1199051)2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

times int

1199052

1199051

1198641003817100381710038171003817T(1199052 minus 119904)120590(119904 119909(119904))

1003817100381710038171003817

2119889119904

(25)

Hence using the strong continuity ofT(119905) and Lebesguersquosdominated convergence theorem we conclude that the right-hand side of the previous inequalities tends to zero as 119905

2minus

1199051rarr 0 Thus we conclude 119865

120572(119909)(119905) is continuous from

the right of [0 119887) A similar argument shows that it is alsocontinuous from the left of (0 119887]

Theorem 13 Assume hypotheses (i) and (ii) are satisfiedThenthe system (1) has a mild solution on 119869

Proof We prove the existence of a fixed point of the operator119865120572by using the contractionmapping principle First we show

that 119865120572(1198672) sub 119867

2 Let 119909 isin 119867

2 From (20) we obtain

1198641003817100381710038171003817119865120572119909(119905)

1003817100381710038171003817

2le 4[sup

119905isin119869

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2] (26)

Using assumptions (i)-(ii) Lemma 11 and standard com-putations yields

sup119905isin119869

1198641003817100381710038171003817Π119909

1(119905)1003817100381710038171003817

2le 11986221198722

01198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)] (27)

sup119905isin119869

4

sum

119894=2

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2

le (1198621198720

Γ(119902))

21198872119902minus1

2119902 minus 11198612 (

1

119887+ 1199092

1198672

)

+ (1198621198720

Γ (119902))

2

[1198872119902minus1

2119902 minus 11198732 minus

1198872119902minus1

2119902 minus 11198712119871120590] (1 + 119909

2

1198672

)

(28)

Hence (26)ndash(28) imply that 11986410038171003817100381710038171198651205721199091003817100381710038171003817

2

1198672

lt infin ByLemma 12 119865

120572119909 isin 119867

2 Thus for each 120572 gt 0 the operator

119865120572maps 119867

2into itself Next we use the Banach fixed point

theorem to prove that 119865120572has a unique fixed point in119867

2 We

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

conditions Moreover there exist positive constants1198731 1198732 gt 0 1198711 1198712 gt 0 and 1198961 1198962 gt 0 such that

1003817100381710038171003817119891(119905 119909) minus 119891(119905 119910)1003817100381710038171003817

2le 1198731

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119891(119905 119909)1003817100381710038171003817

2le 1198732(1 + 119909

2)

1003817100381710038171003817120590(119905 119909) minus 120590(119905 119910)1003817100381710038171003817

2

1198710

2

le 1198711

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

120590(119905 119909)2

1198710

2

le 1198712(1 + 119909

2)

1003817100381710038171003817119892(119909) minus 119892(119910)1003817100381710038171003817

2le 1198961

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1003817100381710038171003817119892(119909)1003817100381710038171003817

2le 1198962 (1 + 119909

2)

(13)

(iii) The linear stochastic system is approximately control-lable on 119869

For each 0 le 119905 lt 119887 the operator 120572(120572119868 + Ψ1198870)minus1rarr 0 in

the strong operator topology as 120572 rarr 0+ where Ψ119887

0= int119887

0(119887 minus

119904)2(119902minus1)

T(119887minus119904)119861119861lowastTlowast(119887minus119904)119889119904 is the controllability GramianHere 119861lowast denotes the adjoint of 119861 andTlowast(119905) is the adjoint ofT(119905)

Observe that Sobolev type nonlocal linear fractionaldeterministic control system

119862119863119902

119905[119871119909 (119905)] = 119872119909 (119905) + 119861119906 (119905) 119905 isin 119869

119909 (0) + 119892 (119909 (119905)) = 1199090

(14)

corresponding to (1) is approximately controllable on 119869 iffthe operator 120572(120572119868 + Ψ119887

0)minus1rarr 0 strongly as 120572 rarr 0

+ Theapproximate controllability for linear fractional deterministiccontrol system (14) is a natural generalization of approximatecontrollability of linear first order control system (119902 = 1119892 = 0 and 119871 is the identity) [50]

Definition 9 The system (1) is approximately controllable on119869 ifR(119887) = 1198712(Ω Γ

119887 119883) where

R (119887) = 119909 (119887) = 119909 (119887 119906) 119906 isin 1198712

Γ(119869 119880) (15)

Here 1198712Γ(119869 119880) is the closed subspace of 1198712

Γ(119869 times Ω119880)

consisting of all Γ119905-adapted 119880-valued stochastic processes

The following lemma is required to define the controlfunction [35]

Lemma 10 For any 119909119887isin 1198712(Γ119887 119883) there exists 120593 isin

1198712

Γ(Ω 1198712(0 119887 119871

0

2)) such that 119909

119887= 119864119909119887+ int119887

0120593(119904)119889119908(119904)

Now for any 120572 gt 0 and 119909119887isin 1198712(Γ119887 119883) one defines the

control function in the following form

119906120572(119905 119909) = 119861

lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times [(120572119868 + Ψ119887

0)minus1

119864119909119887 minusS (119887) 119871 [1199090 minus 119892 (119909)]

+int

119905

0

(120572119868 + Ψ119887

0)minus1

120593 (119904) 119889119908 (119904)]

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909 (119904)) 119889119904

minus 119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

(16)

Lemma 11 There exist positive real constants such thatfor all 119909 119910 isin 119867

2 one has

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2le 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (17)

1198641003817100381710038171003817119906120572(119905 119909)

1003817100381710038171003817

2le (

1

119887+ 119864119909(119905)

2) (18)

Proof We start to prove (17) Let 119909 119910 isin 1198672 from Holderrsquos

inequality Lemma 6 and the assumption on the data weobtain

1198641003817100381710038171003817119906120572(119905 119909) minus 119906

120572(119905 119910)

1003817100381710038171003817

2

le 3119864100381710038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905) (120572119868 + Ψ

119887

0)minus1

timesS (119887) 119871 [119892 (119909 (119905)) minus 119892 (119910 (119905))]100381710038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [119891 (119904 119909 (119904)) minus 119891 (119904 119910 (119904))] 11988911990410038171003817100381710038171003817

2

+ 311986410038171003817100381710038171003817119861lowast(119887 minus 119905)

119902minus1Tlowast(119887 minus 119905)

times int

119905

0

(120572119868 + Ψ119887

0)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904)

times [120590 (119904 119909 (119904)) minus 120590 (119904 119910 (119904))] 119889119908 (119904)10038171003817100381710038171003817

2

Abstract and Applied Analysis 5

le3

12057221198612(119887)2119902minus2(1198621198720)

2(1198621198720

Γ(119902))

2

119871211989611198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198731

times int

119905

0

1198641003817100381710038171003817119909 (119904) minus 119910 (119904)

1003817100381710038171003817

2119889119904

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198711

times int

119905

0

1198641003817100381710038171003817119909(119904) minus 119910(119904)

1003817100381710038171003817

2119889119904

le 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(19)

where = (31205722)1198612(119887)2119902minus2(119862119872

0)2(1198621198720Γ(119902))

211987121198961+

(1198621198720Γ(119902))

4(1198872119902minus1(2119902 minus 1))119887[119873

1+ 1198711] The proof of the

inequality (18) can be established in a similar way to that of(17)

3 Approximate Controllability

In this section we formulate and prove conditions forthe existence and approximate controllability results of thenonlocal fractional stochastic dynamic control system ofSobolev type (1) using the contractionmapping principle Forany 120572 gt 0 define the operator 119865120572 1198672 rarr 1198672 by

119865120572119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906120572(119904 119909) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(20)

We state and prove the following lemma which will beused for the main results

Lemma 12 For any 119909 isin 1198672 119865120572(119909)(119905) is continuous on 119869 in

1198712-sense

Proof Let 0 le 1199051lt 1199052le 119887 Then for any fixed 119909 isin 119867

2 from

(20) we have

1198641003817100381710038171003817(119865120572119909)(1199052) minus (119865120572119909)(1199051)

1003817100381710038171003817

2le 4[

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(1199052) minus Π119909

119894(1199051)1003817100381710038171003817

2]

(21)

We begin with the first term and get

1198641003817100381710038171003817Π119909

1(1199052) minus Π

119909

1(1199051)1003817100381710038171003817

2= 1198641003817100381710038171003817(S(1199052) minus S(1199051))119871[1199090 minus 119892(119909)]

1003817100381710038171003817

2

le 1198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)]

times 1198641003817100381710038171003817S(1199052) minusS(1199051)

1003817100381710038171003817

2

(22)

The strong continuity of S(119905) implies that the right-handside of the last inequality tends to zero as 119905

2minus 1199051rarr 0

Next it follows fromHolderrsquos inequality and assumptionson the data that

1198641003817100381710038171003817Π119909

2(1199052) minus Π

119909

2(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052 minus 119904)119902minus1

T (1199052 minus 119904) 119861119906120572(119904 119909) 119889119904

minusint

1199051

0

(1199051 minus 119904)119902minus1

T(1199051 minus 119904)119861119906120572(119904 119909)119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051 minus 119904)119902minus1(T (1199052 minus 119904) minusT (1199051 minus 119904))

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)T (119905

2minus 119904)

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T(1199052minus 119904)119861119906

120572(119904 119909)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))119861119906

120572(119904 119909)119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

1198612(int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

1198612int

1199052

1199051

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904

(23)

Also we have

1198641003817100381710038171003817Π119909

3(1199052) minus Π119909

3(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

minusint

1199051

0

(1199051minus 119904)119902minus1

T(1199051minus 119904)119891(119904 119909(119904))119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

6 Abstract and Applied Analysis

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)

timesT (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891(119904 119909(119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T (1199052 minus 119904) minusT (1199051 minus 119904)) 119891 (119904 119909 (119904)) 119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

int

1199052

1199051

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904

(24)

Furthermore we use Lemma6 andprevious assumptionswe obtain

1198641003817100381710038171003817Π119909

4(1199052) minus Π119909

4(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

minusint

1199051

0

(1199051minus 119904)119902minus1

T (1199051minus 119904) 120590 (119904 119909 (119904)) 119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)T (1199052 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052 minus 119904)119902minus1

T(1199052 minus 119904)120590(119904 119909(119904))119889119908(119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 119871120590

1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))120590(119904 119909(119904))119889119904

1003817100381710038171003817

2

+ 119871120590(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817T (1199052 minus 119904) 120590 (119904 119909 (119904))

1003817100381710038171003817

2119889119904)

+ 119871120590

(1199052 minus 1199051)2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

times int

1199052

1199051

1198641003817100381710038171003817T(1199052 minus 119904)120590(119904 119909(119904))

1003817100381710038171003817

2119889119904

(25)

Hence using the strong continuity ofT(119905) and Lebesguersquosdominated convergence theorem we conclude that the right-hand side of the previous inequalities tends to zero as 119905

2minus

1199051rarr 0 Thus we conclude 119865

120572(119909)(119905) is continuous from

the right of [0 119887) A similar argument shows that it is alsocontinuous from the left of (0 119887]

Theorem 13 Assume hypotheses (i) and (ii) are satisfiedThenthe system (1) has a mild solution on 119869

Proof We prove the existence of a fixed point of the operator119865120572by using the contractionmapping principle First we show

that 119865120572(1198672) sub 119867

2 Let 119909 isin 119867

2 From (20) we obtain

1198641003817100381710038171003817119865120572119909(119905)

1003817100381710038171003817

2le 4[sup

119905isin119869

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2] (26)

Using assumptions (i)-(ii) Lemma 11 and standard com-putations yields

sup119905isin119869

1198641003817100381710038171003817Π119909

1(119905)1003817100381710038171003817

2le 11986221198722

01198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)] (27)

sup119905isin119869

4

sum

119894=2

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2

le (1198621198720

Γ(119902))

21198872119902minus1

2119902 minus 11198612 (

1

119887+ 1199092

1198672

)

+ (1198621198720

Γ (119902))

2

[1198872119902minus1

2119902 minus 11198732 minus

1198872119902minus1

2119902 minus 11198712119871120590] (1 + 119909

2

1198672

)

(28)

Hence (26)ndash(28) imply that 11986410038171003817100381710038171198651205721199091003817100381710038171003817

2

1198672

lt infin ByLemma 12 119865

120572119909 isin 119867

2 Thus for each 120572 gt 0 the operator

119865120572maps 119867

2into itself Next we use the Banach fixed point

theorem to prove that 119865120572has a unique fixed point in119867

2 We

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

le3

12057221198612(119887)2119902minus2(1198621198720)

2(1198621198720

Γ(119902))

2

119871211989611198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198731

times int

119905

0

1198641003817100381710038171003817119909 (119904) minus 119910 (119904)

1003817100381710038171003817

2119889119904

+3

12057221198612(119887)2119902minus2(1198621198720

Γ (119902))

4

1198872119902minus1

(2119902 minus 1)1198711

times int

119905

0

1198641003817100381710038171003817119909(119904) minus 119910(119904)

1003817100381710038171003817

2119889119904

le 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(19)

where = (31205722)1198612(119887)2119902minus2(119862119872

0)2(1198621198720Γ(119902))

211987121198961+

(1198621198720Γ(119902))

4(1198872119902minus1(2119902 minus 1))119887[119873

1+ 1198711] The proof of the

inequality (18) can be established in a similar way to that of(17)

3 Approximate Controllability

In this section we formulate and prove conditions forthe existence and approximate controllability results of thenonlocal fractional stochastic dynamic control system ofSobolev type (1) using the contractionmapping principle Forany 120572 gt 0 define the operator 119865120572 1198672 rarr 1198672 by

119865120572119909 (119905) = S (119905) 119871 [1199090 minus 119892 (119909)]

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) [119861119906120572(119904 119909) + 119891 (119904 119909 (119904))] 119889119904

+ int

119905

0

(119905 minus 119904)119902minus1

T (119905 minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

(20)

We state and prove the following lemma which will beused for the main results

Lemma 12 For any 119909 isin 1198672 119865120572(119909)(119905) is continuous on 119869 in

1198712-sense

Proof Let 0 le 1199051lt 1199052le 119887 Then for any fixed 119909 isin 119867

2 from

(20) we have

1198641003817100381710038171003817(119865120572119909)(1199052) minus (119865120572119909)(1199051)

1003817100381710038171003817

2le 4[

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(1199052) minus Π119909

119894(1199051)1003817100381710038171003817

2]

(21)

We begin with the first term and get

1198641003817100381710038171003817Π119909

1(1199052) minus Π

119909

1(1199051)1003817100381710038171003817

2= 1198641003817100381710038171003817(S(1199052) minus S(1199051))119871[1199090 minus 119892(119909)]

1003817100381710038171003817

2

le 1198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)]

times 1198641003817100381710038171003817S(1199052) minusS(1199051)

1003817100381710038171003817

2

(22)

The strong continuity of S(119905) implies that the right-handside of the last inequality tends to zero as 119905

2minus 1199051rarr 0

Next it follows fromHolderrsquos inequality and assumptionson the data that

1198641003817100381710038171003817Π119909

2(1199052) minus Π

119909

2(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052 minus 119904)119902minus1

T (1199052 minus 119904) 119861119906120572(119904 119909) 119889119904

minusint

1199051

0

(1199051 minus 119904)119902minus1

T(1199051 minus 119904)119861119906120572(119904 119909)119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051 minus 119904)119902minus1(T (1199052 minus 119904) minusT (1199051 minus 119904))

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)T (119905

2minus 119904)

times 119861119906120572(119904 119909) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T(1199052minus 119904)119861119906

120572(119904 119909)119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))119861119906

120572(119904 119909)119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

1198612(int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

1198612int

1199052

1199051

1198641003817100381710038171003817119906120572(119904 119909)

1003817100381710038171003817

2119889119904

(23)

Also we have

1198641003817100381710038171003817Π119909

3(1199052) minus Π119909

3(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

minusint

1199051

0

(1199051minus 119904)119902minus1

T(1199051minus 119904)119891(119904 119909(119904))119889119904

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

6 Abstract and Applied Analysis

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)

timesT (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891(119904 119909(119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T (1199052 minus 119904) minusT (1199051 minus 119904)) 119891 (119904 119909 (119904)) 119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

int

1199052

1199051

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904

(24)

Furthermore we use Lemma6 andprevious assumptionswe obtain

1198641003817100381710038171003817Π119909

4(1199052) minus Π119909

4(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

minusint

1199051

0

(1199051minus 119904)119902minus1

T (1199051minus 119904) 120590 (119904 119909 (119904)) 119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)T (1199052 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052 minus 119904)119902minus1

T(1199052 minus 119904)120590(119904 119909(119904))119889119908(119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 119871120590

1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))120590(119904 119909(119904))119889119904

1003817100381710038171003817

2

+ 119871120590(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817T (1199052 minus 119904) 120590 (119904 119909 (119904))

1003817100381710038171003817

2119889119904)

+ 119871120590

(1199052 minus 1199051)2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

times int

1199052

1199051

1198641003817100381710038171003817T(1199052 minus 119904)120590(119904 119909(119904))

1003817100381710038171003817

2119889119904

(25)

Hence using the strong continuity ofT(119905) and Lebesguersquosdominated convergence theorem we conclude that the right-hand side of the previous inequalities tends to zero as 119905

2minus

1199051rarr 0 Thus we conclude 119865

120572(119909)(119905) is continuous from

the right of [0 119887) A similar argument shows that it is alsocontinuous from the left of (0 119887]

Theorem 13 Assume hypotheses (i) and (ii) are satisfiedThenthe system (1) has a mild solution on 119869

Proof We prove the existence of a fixed point of the operator119865120572by using the contractionmapping principle First we show

that 119865120572(1198672) sub 119867

2 Let 119909 isin 119867

2 From (20) we obtain

1198641003817100381710038171003817119865120572119909(119905)

1003817100381710038171003817

2le 4[sup

119905isin119869

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2] (26)

Using assumptions (i)-(ii) Lemma 11 and standard com-putations yields

sup119905isin119869

1198641003817100381710038171003817Π119909

1(119905)1003817100381710038171003817

2le 11986221198722

01198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)] (27)

sup119905isin119869

4

sum

119894=2

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2

le (1198621198720

Γ(119902))

21198872119902minus1

2119902 minus 11198612 (

1

119887+ 1199092

1198672

)

+ (1198621198720

Γ (119902))

2

[1198872119902minus1

2119902 minus 11198732 minus

1198872119902minus1

2119902 minus 11198712119871120590] (1 + 119909

2

1198672

)

(28)

Hence (26)ndash(28) imply that 11986410038171003817100381710038171198651205721199091003817100381710038171003817

2

1198672

lt infin ByLemma 12 119865

120572119909 isin 119867

2 Thus for each 120572 gt 0 the operator

119865120572maps 119867

2into itself Next we use the Banach fixed point

theorem to prove that 119865120572has a unique fixed point in119867

2 We

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)

timesT (1199052minus 119904) 119891 (119904 119909 (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052minus 119904)119902minus1

T (1199052minus 119904) 119891(119904 119909(119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

le1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T (1199052 minus 119904) minusT (1199051 minus 119904)) 119891 (119904 119909 (119904)) 119889119904

1003817100381710038171003817

2

+ (1198621198720

Γ(119902))

2

(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904)

+(1199052 minus 1199051)

2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

int

1199052

1199051

1198641003817100381710038171003817119891(119904 119909(119904))

1003817100381710038171003817

2119889119904

(24)

Furthermore we use Lemma6 andprevious assumptionswe obtain

1198641003817100381710038171003817Π119909

4(1199052) minus Π119909

4(1199051)1003817100381710038171003817

2

= 119864

10038171003817100381710038171003817100381710038171003817

int

1199052

0

(1199052minus 119904)119902minus1

T (1199052minus 119904) 120590 (119904 119909 (119904)) 119889119908 (119904)

minusint

1199051

0

(1199051minus 119904)119902minus1

T (1199051minus 119904) 120590 (119904 119909 (119904)) 119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2

le 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

(1199051minus 119904)119902minus1(T (1199052minus 119904) minusT (119905

1minus 119904))

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

10038171003817100381710038171003817100381710038171003817

int

1199051

0

((1199052 minus 119904)119902minus1minus (1199051 minus 119904)

119902minus1)T (1199052 minus 119904)

times 120590 (119904 119909 (119904)) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 119864

100381710038171003817100381710038171003817100381710038171003817

int

1199052

1199051

(1199052 minus 119904)119902minus1

T(1199052 minus 119904)120590(119904 119909(119904))119889119908(119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 119871120590

1199052119902minus1

1

2119902 minus 1

times int

1199051

0

1198641003817100381710038171003817(T(1199052 minus 119904) minusT(1199051 minus 119904))120590(119904 119909(119904))119889119904

1003817100381710038171003817

2

+ 119871120590(int

1199051

0

((1199052minus 119904)119902minus1minus (1199051minus 119904)119902minus1)2

119889119904)

times (int

1199051

0

1198641003817100381710038171003817T (1199052 minus 119904) 120590 (119904 119909 (119904))

1003817100381710038171003817

2119889119904)

+ 119871120590

(1199052 minus 1199051)2119902minus1

1 minus 2119902(1198621198720

Γ(119902))

2

times int

1199052

1199051

1198641003817100381710038171003817T(1199052 minus 119904)120590(119904 119909(119904))

1003817100381710038171003817

2119889119904

(25)

Hence using the strong continuity ofT(119905) and Lebesguersquosdominated convergence theorem we conclude that the right-hand side of the previous inequalities tends to zero as 119905

2minus

1199051rarr 0 Thus we conclude 119865

120572(119909)(119905) is continuous from

the right of [0 119887) A similar argument shows that it is alsocontinuous from the left of (0 119887]

Theorem 13 Assume hypotheses (i) and (ii) are satisfiedThenthe system (1) has a mild solution on 119869

Proof We prove the existence of a fixed point of the operator119865120572by using the contractionmapping principle First we show

that 119865120572(1198672) sub 119867

2 Let 119909 isin 119867

2 From (20) we obtain

1198641003817100381710038171003817119865120572119909(119905)

1003817100381710038171003817

2le 4[sup

119905isin119869

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2] (26)

Using assumptions (i)-(ii) Lemma 11 and standard com-putations yields

sup119905isin119869

1198641003817100381710038171003817Π119909

1(119905)1003817100381710038171003817

2le 11986221198722

01198712[100381710038171003817100381711990901003817100381710038171003817

2+ 1198962(1 + 119909

2)] (27)

sup119905isin119869

4

sum

119894=2

1198641003817100381710038171003817Π119909

119894(119905)1003817100381710038171003817

2

le (1198621198720

Γ(119902))

21198872119902minus1

2119902 minus 11198612 (

1

119887+ 1199092

1198672

)

+ (1198621198720

Γ (119902))

2

[1198872119902minus1

2119902 minus 11198732 minus

1198872119902minus1

2119902 minus 11198712119871120590] (1 + 119909

2

1198672

)

(28)

Hence (26)ndash(28) imply that 11986410038171003817100381710038171198651205721199091003817100381710038171003817

2

1198672

lt infin ByLemma 12 119865

120572119909 isin 119867

2 Thus for each 120572 gt 0 the operator

119865120572maps 119867

2into itself Next we use the Banach fixed point

theorem to prove that 119865120572has a unique fixed point in119867

2 We

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

claim that there exists a natural 119899 such that119865119899120572is a contraction

on1198672 Indeed let 119909 119910 isin 119867

2 we have

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2

le 4

4

sum

119894=1

1198641003817100381710038171003817Π119909

119894(119905) minus Π

119910

119894(119905)1003817100381710038171003817

2

le 4119896111986221198722

011987121198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

+ 4(1198621198720

Γ (119902))

2

times [1198612 1198872119902minus1

2119902 minus 1+1198872119902minus1

2119902 minus 11198731+1198872119902minus1

2119902 minus 11198711119871120590]

times 1198641003817100381710038171003817119909(119905) minus 119910(119905)

1003817100381710038171003817

2

(29)

Hence we obtain a positive real constant 120574(120572) such that

1198641003817100381710038171003817(119865120572119909) (119905) minus (119865120572119910) (119905)

1003817100381710038171003817

2le 120574 (120572) 119864

1003817100381710038171003817119909(119905) minus 119910(119905)1003817100381710038171003817

2 (30)

for all 119905 isin 119869 and all 119909 119910 isin 1198672 For any natural number

119899 it follows from the successive iteration of the previousinequality (30) that by taking the supremum over 119869

1003817100381710038171003817(119865119899

120572119909)(119905) minus (119865

119899

120572119910)(119905)

1003817100381710038171003817

2

1198672

le120574119899(120572)

119899

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

1198672

(31)

For any fixed 120572 gt 0 for sufficiently large 119899 120574119899(120572)119899 lt 1It follows from (31) that 119865119899

120572is a contraction mapping so that

the contraction principle ensures that the operator 119865120572has a

unique fixed point 119909120572 in 1198672 which is a mild solution of (1)

Theorem 14 Assume that the assumptions (i)ndash(iii) holdFurther if the functions 119891 and 120590 are uniformly bounded andT(119905) 119905 ge 0 is compact then the system (1) is approximatelycontrollable on 119869

Proof Let 119909120572be a fixed point of 119865

120572 By using the stochastic

Fubini theorem it can be easily seen that

119909120572 (119887)

= 119909119887minus 120572(120572119868 + Ψ)

minus1(119864119909119887minusS (119887) 119871 [1199090 minus 119892 (119909)])

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

(119887 minus 119904)119902minus1

T (119887 minus 119904) 119891 (119904 119909120572 (119904)) 119889119904

+ 120572int

119887

0

(120572119868 + Ψ119887

119904)minus1

[(119887 minus 119904)119902minus1

T (119887 minus 119904) 120590

times (119904 119909120572 (119904)) minus 120593 (119904)] 119889119908 (119904)

(32)

It follows from the assumption on 119891 119892 and 120590 that thereexists119863 gt 0 such that

1003817100381710038171003817119891(119904 119909120572(119904))1003817100381710038171003817

2+1003817100381710038171003817119892(119909120572(119904))

1003817100381710038171003817

2+1003817100381710038171003817120590(119904 119909120572(119904))

1003817100381710038171003817

2le 119863 (33)

for all 119904 isin 119869 Then there is a subsequence still denoted by119891(119904 119909120572(119904)) 119892(119909120572(119904)) 120590(119904 119909120572(119904)) which converges weakly tosome 119891(119904) 119892(119904) 120590(119904) in 1198842 times 1198710

2

From the previous equation we have

1198641003817100381710038171003817119909120572(119887) minus 119909119887

1003817100381710038171003817

2

le 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1

(119864119909119887minusS(119887)119871119909

0)100381710038171003817100381710038171003817

2

+ 8119864100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus1100381710038171003817100381710038171003817

21003817100381710038171003817S (119887) 119871 (119892 (119909120572 (119904)) minus 119892 (119904))

1003817100381710038171003817

2

+ 811986410038171003817100381710038171003817120572(120572119868 + Ψ

119887

0)minus110038171003817100381710038171003817

21003817100381710038171003817S(119887)119871119892(119904)1003817100381710038171003817

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

120593(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T (119887 minus 119904) (119891 (119904 119909120572 (119904)) minus 119891 (119904))

1003817100381710038171003817 119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)119891(119904)100381710038171003817100381710038171003817119889119904)

2

+ 8119864(int

119887

0

(119887 minus 119904)119902minus1100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1100381710038171003817100381710038171003817

times1003817100381710038171003817T(119887 minus 119904)(120590(119904 119909120572(119904)) minus 120590(119904))

1003817100381710038171003817

2

1198710

2

119889119904)

+ 8119864int

119887

0

(119887 minus 119904)119902minus1

times100381710038171003817100381710038171003817120572(120572119868 + Ψ

119887

119904)minus1

T(119887 minus 119904)120590(119904)100381710038171003817100381710038171003817

2

1198710

2

119889119904

(34)

On the other hand by assumption (iii) for all 0 le 119904 lt119887 the operator 120572(120572119868 + Ψ119887

119904)minus1rarr 0 strongly as 120572 rarr 0

+

and moreover 120572(120572119868 + Ψ119887119904)minus1 le 1 Thus by the Lebesgue

dominated convergence theorem and the compactness ofboth S(119905) andT(119905) it is implied that 119864119909

120572(119887) minus 119909

1198872rarr 0 as

120572 rarr 0+ Hence we conclude the approximate controllability

of (1)

In order to illustrate the abstract results of this work wegive the following example

4 Example

Consider the fractional stochastic system with nonlocalcondition of Sobolev type

120597119902

120597119905119902[119909 (119911 119905) minus 119909119911119911 (119911 119905)] minus

1205972

1205971199112119909 (119911 119905)

= 120583 (119911 119905) + 119891 (119905 119909 (119911 119905)) + (119905 119909 (119911 119905))119889119908 (119905)

119889119905

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

119909 (119911 0) +

119898

sum

119896=1

119888119896119909 (119911 119905119896) = 1199090 (119911) 119911 isin [0 1]

119909 (0 119905) = 119909 (1 119905) = 0 119905 isin 119869

(35)

where 0 lt 119902 le 1 0 lt 1199051lt sdot sdot sdot lt 119905

119898lt 119887 and 119888

119896are positive

constants 119896 = 1 119898 the functions 119909(119905)(119911) = 119909(119905 119911)119891(119905 119909(119905))(119911) = 119891(119905 119909(119905 119911)) 120590(119905 119909(119905))(119911) = (119905 119909(119905 119911)) and119892(119909(119905))(119911) = sum

119898

119896=1119888119896119909(119911 119905119896) The bounded linear operator

119861 119880 rarr 119883 is defined by119861119906(119905)(119911) = 120583(119911 119905) 0 le 119911 le 1 119906 isin 119880119908(119905) is a two-sided and standard one-dimensional Brownianmotion defined on the filtered probability space (Ω Γ 119875)

Let119883 = 119864 = 119880 = 1198712[0 1] define the operators 119871 119863(119871) sub119883 rarr 119884 and 119872 119863(119872) sub 119883 rarr 119884 by 119871119909 = 119909 minus 11990910158401015840 and119872119909 = minus119909

10158401015840 where domains119863(119871) and119863(119872) are given by

119909 isin 119883 119909 1199091015840 are absolutely continuous 11990910158401015840 isin 119883

119909 (0) = 119909 (1) = 0

(36)

Then 119871 and119872 can be written respectively as

119871119909 =

infin

sum

119899=1

(1 + 1198992) (119909 119909

119899) 119909119899 119909 isin 119863 (119871)

119872119909 =

infin

sum

119899=1

minus1198992(119909 119909119899) 119909119899 119909 isin 119863 (119872)

(37)

where 119909119899(119911) = (radic2120587) sin 119899119911 119899 = 1 2 is the orthogonal

set of eigen functions of119872 Further for any 119909 isin 119883 we have

119871minus1119909 =

infin

sum

119899=1

1

1 + 1198992(119909 119909119899) 119909119899

119872119871minus1119909 =

infin

sum

119899=1

minus1198992

1 + 1198992(119909 119909119899) 119909119899

119878 (119905) 119909 =

infin

sum

119899=1

exp( minus1198992119905

1 + 1198992) (119909 119909

119899) 119909119899

(38)

It is easy to see that 119871minus1 is compact and bounded with119871minus1 le 1 and119872119871minus1 generates the above strongly continuous

semigroup 119878(119905) on119884with 119878(119905) le 119890minus119905 le 1Therefore with theabove choices the system (35) can be written as an abstractformulation of (1) and thus Theorem 13 can be applied toguarantee the existence of mild solution of (35) Moreoverit can be easily seen that Sobolev type deterministic linearfractional control system corresponding to (35) is approxi-mately controllable on 119869 which means that all conditions ofTheorem 14 are satisfied Thus fractional stochastic controlsystem of Sobolev type (35) is approximately controllable on119869

5 Conclusion

Sufficient conditions for the approximate controllability ofa class of dynamic control systems described by Sobolev

type nonlocal fractional stochastic differential equations inHilbert spaces are considered Using fixed point techniquefractional calculations stochastic analysis and methodsadopted directly from deterministic control problems Inparticular conditions are formulated and proved underthe assumption that the approximate controllability of thestochastic control nonlinear dynamical system is implied bythe approximate controllability of its corresponding linearpart More precisely the controllability problem is trans-formed into a fixed point problem for an appropriate non-linear operator in a function space The main used tools arethe above required conditions we guarantee the existence ofa fixed point of this operator and study controllability of theconsidered systems

Degenerate stochastic differential equations model thephenomenon of convection-diffusion of ideal fluids andtherefore arise in a wide variety of important applicationsincluding for instance two or three phase flows in porousmedia or sedimentation-consolidation processes Howeverto the best of our knowledge no results yet exist onapproximate controllability for fractional stochastic degener-ate systems Upon making some appropriate assumptions byemploying the ideas and techniques as in this paper one canestablish the approximate controllability results for a class offractional stochastic degenerate differential equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of ComplexityNonlinearity and Chaos World Scientific 2012

[2] M Bragdi A Debbouche and D Baleanu ldquoExistence ofsolutions for fractional differential inclusions with separatedboundary conditions in Banach spacerdquo Advances in Mathemat-ical Physics vol 2013 Article ID 426061 5 pages 2013

[3] A Debbouche ldquoFractional nonlocal impulsive quasilinearmulti-delay integro-differential systemsrdquoAdvances inDifferenceEquations vol 2011 article 5 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalNonlinear Integro-Differential Equations of Fractional OrdersrdquoBoundary Value Problems no 2012 article 78 2012

[5] A M A El-Sayed ldquoFractional-order diffusion-wave equationrdquoInternational Journal ofTheoretical Physics vol 35 no 2 pp 311ndash322 1996

[6] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics vol 378 pp 291ndash348Springer New York NY USA 1997

[7] A B Malinowska and D F M Torres Introduction to the Frac-tional Calculus of Variations Imperial College Press LondonUK 2012

[8] N Nyamoradi D Baleanu and T Bashiri ldquoPositive solutions tofractional boundary value problems with nonlinear boundaryconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID579740 20 pages 2013

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

[9] B Ahmad and S K Ntouyas ldquoA note on fractional differen-tial equations with fractional separated boundary conditionsrdquoAbstract and Applied Analysis vol 2012 Article ID 818703 11pages 2012

[10] R Sakthivel R Ganesh and S Suganya ldquoApproximate con-trollability of fractional neutral stochastic system with infinitedelayrdquo Reports on Mathematical Physics vol 70 no 3 pp 291ndash311 2012

[11] R Sakthivel R Ganesh Y Ren and S M Anthoni ldquoApproxi-mate controllability of nonlinear fractional dynamical systemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 12 pp 3498ndash3508 2013

[12] A E Bashirov and N I Mahmudov ldquoOn concepts of control-lability for deterministic and stochastic systemsrdquo SIAM Journalon Control and Optimization vol 37 no 6 pp 1808ndash1821 1999

[13] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1442ndash1450 2011

[14] A Debbouche and D Baleanu ldquoExact null controllability forfractional nonlocal integrodifferential equations via implicitevolution systemrdquo Journal of Applied Mathematics vol 2012Article ID 931975 17 pages 2012

[15] R Sakthivel N I Mahmudov and J J Nieto ldquoControllabilityfor a class of fractional-order neutral evolution control systemsrdquoApplied Mathematics and Computation vol 218 no 20 pp10334ndash10340 2012

[16] R Triggiani ldquoA note on the lack of exact controllability formild solutions in Banach spacesrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 407ndash411 1977

[17] J P Dauer and N I Mahmudov ldquoApproximate controllabilityof semilinear functional equations in Hilbert spacesrdquo Journal ofMathematical Analysis and Applications vol 273 no 2 pp 310ndash327 2002

[18] N Sukavanam and N K Tomar ldquoApproximate controllabilityof semilinear delay control systemsrdquo Nonlinear FunctionalAnalysis and Applications vol 12 no 1 pp 53ndash59 2007

[19] W Bian ldquoApproximate controllability for semilinear systemsrdquoActa Mathematica Hungarica vol 81 no 1-2 pp 41ndash57 1998

[20] E N Chukwu and S M Lenhart ldquoControllability questions fornonlinear systems in abstract spacesrdquo Journal of OptimizationTheory and Applications vol 68 no 3 pp 437ndash462 1991

[21] J-M Jeong J-R Kim and H-H Roh ldquoControllability forsemilinear retarded control systems in Hilbert spacesrdquo Journalof Dynamical and Control Systems vol 13 no 4 pp 577ndash5912007

[22] J-M Jeong and H-G Kim ldquoControllability for semilinearfunctional integrodifferential equationsrdquo Bulletin of the KoreanMathematical Society vol 46 no 3 pp 463ndash475 2009

[23] N Sukavanam and Divya ldquoApproximate controllability ofabstract semilinear deterministic control systemrdquo Bulletin of theCalcutta Mathematical Society vol 96 no 3 pp 195ndash202 2004

[24] S Kumar and N Sukavanam ldquoApproximate controllabilityof fractional order semilinear systems with bounded delayrdquoJournal of Differential Equations vol 252 no 11 pp 6163ndash61742012

[25] S Rathinasamy and R Yong ldquoApproximate controllability offractional differential equations with state-dependent delayrdquoResults in Mathematics vol 63 no 3-4 pp 949ndash963 2013

[26] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquo

Computers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[27] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[28] J Cao Q Yang Z Huang and Q Liu ldquoAsymptotically almostperiodic solutions of stochastic functional differential equa-tionsrdquo Applied Mathematics and Computation vol 218 no 5pp 1499ndash1511 2011

[29] J Cao Q Yang and Z Huang ldquoOn almost periodic mild solu-tions for stochastic functional differential equationsrdquoNonlinearAnalysis Real World Applications vol 13 no 1 pp 275ndash2862012

[30] Y-K Chang Z-H Zhao G M NrsquoGuerekata and R MaldquoStepanov-like almost automorphy for stochastic processesand applications to stochastic differential equationsrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1130ndash11392011

[31] X Mao Stochastic Differential Equations and Applications EllisHorwood Chichester UK 1997

[32] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[33] N I Mahmudov ldquoApproximate controllability of semilineardeterministic and stochastic evolution equations in abstractspacesrdquo SIAM Journal on Control and Optimization vol 42 no5 pp 1604ndash1622 2003

[34] F Li J Liang and H K Xu ldquoExistence of mild solutions forfractional integrodifferential equations of Sobolev type withnonlocal conditionsrdquo Journal of Mathematical Analysis andApplications vol 391 pp 510ndash525 2012

[35] R Sakthivel S Suganya and S M Anthoni ldquoApproximatecontrollability of fractional stochastic evolution equationsrdquoComputers amp Mathematics with Applications vol 63 no 3 pp660ndash668 2012

[36] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[37] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[38] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[39] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[40] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Technical University ofKosice Kosice Slovak Republic 1999

[41] E Hille and R S Phillips Functional Analysis and Semi-Groups vol 31 of American Mathematical Society ColloquiumPublications American Mathematical Society Providence RIUSA 1957

[42] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations Springer New York NY USA1983

[43] S D Zaidman Abstract Differential Equations PitmanAdvanced Publishing Program San Francisco Calif USA1979

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[44] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 Cambridge University Press CambridgeUK 1992

[45] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers amp Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[46] A Debbouche and M M El-Borai ldquoWeak almost periodicand optimal mild solutions of fractional evolution equationsrdquoElectronic Journal of Differential Equations no 46 pp 1ndash8 2009

[47] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons andFractals vol 14 no 3 pp 433ndash440 2002

[48] Z Yan ldquoApproximate controllability of partial neutral func-tional differential systems of fractional order with state-dependent delayrdquo International Journal of Control vol 85 no8 pp 1051ndash1062 2012

[49] Z Yan ldquoApproximate controllability of fractional neutralintegro-differental inclusions with state-dependent delay inHilbert spacesrdquo IMA Journal of Mathematical Control andInformation 2012

[50] A Debbouche andD FM Torres ldquoApproximate controllabilityof fractional nonlocal delay semilinear systems in Hilbertspacesrdquo International Journal of Control vol 86 no 9 pp 1577ndash1585 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of