Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 567089 10 pageshttpdxdoiorg1011552013567089

Research ArticleReachability and Controllability of Fractional SingularDynamical Systems with Control Delay

Hai Zhang12 Jinde Cao13 and Wei Jiang4

1 Department of Mathematics Southeast University Nanjing Jiangsu 210096 China2Department of Mathematics Anqing Normal University Anqing Anhui 246133 China3Department of Mathematics King Abdulaziz University Jeddah 21589 Saudi Arabia4 School of Mathematical Sciences Anhui University Hefei Anhui 230039 China

Correspondence should be addressed to Jinde Cao jdcaoseueducn

Received 3 August 2013 Revised 25 September 2013 Accepted 25 September 2013

Academic Editor Han H Choi

Copyright copy 2013 Hai Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay Thefactors of such systems including the Caputorsquos fractional derivative control delay and singular coefficient matrix are taken intoaccount synchronouslyThe state structure of fractional singular dynamical systems with control delay is characterized by analysingthe state response and reachable set A set of sufficient and necessary conditions of controllability for such systems are establishedbased on the algebraic approach Moreover an example is provided to illustrate the effectiveness and applicability of the proposedcriteria

1 Introduction

Singular systems play important roles in mathematical mod-eling of real-life problems arising in a wide range of applica-tions Depending on the area of application these models arealso called descriptor systems semistate systems differential-algebraic systems or generalized state-space systems In thepast few decades singular systems with integer-order deriva-tive have been extensively studied due to the fact that singularsystems describe practical systems better than normal ones(see [1ndash8] and references therein)

Fractional calculus and its applications to the sciences andengineering is a recent focus of interest to many researchersFractional differential equations have been proved to be anexcellent tool in the modelling of many phenomena in var-ious fields of engineering physics and economics Actuallyfractional differential equations are considered as alternativemodel to integer differential equations Many practical sys-tems can be represented more accurately through fractionalderivative formulation For more details on fractional calcu-lus theory one can see the monographs of Miller and Ross[9] Podlubny [10] Diethelm [11] and Kilbas et al [12] Frac-tional differential equations involving the Riemann-Liouville

fractional derivative or the Caputo fractional derivative havebeen paid more and more attention in [13ndash18] and referencestherein

On the other hand it is well-known that the issue of con-trollability plays an important role in engineering and controltheory which has close connections to pole assignmentstructural decomposition quadratic optimal control andcontroller design and so on The problem of controllabilityfor various kinds of dynamical systemswith integer derivativehas been extensively studied [1 3ndash5 19ndash22] In particularDai [1] has established the theory of integer-order linearsingular control systems Yip and Sincovec [3] and Tang andLi [4] have investigated the controllability and observabilityof integer-order singular dynamical systems It is worthmentioning that notable contributions have been made tofractional control systems [23ndash35] The different techniqueshave been applied to investigate the controllability of variousfractional dynamical systems such as algebraic method [2324] geometrical analysis [25] fixed point theorems [26ndash30]and semigroup theory [31] Recently Sakthivel et al [32ndash35] have investigated the approximate controllability problemfor various kinds of fractional dynamical systems by usingfixed point techniques which have further enriched and

2 Journal of Applied Mathematics

developed the control theory of fractional dynamical systemsThe earlier studies concerning the controllability of fractionaldynamical systems with control delay can be found in [25 2729] which especially focused on fractional normal systemsHowever it should be emphasized that the control theory offractional singular systems is not yet sufficiently elaboratedcompared to that of fractional normal systems In this regardit is necessary and important to study the controllabilityproblems for fractional singular dynamical systems To thebest of our knowledge there are no relevant reports on reach-ability and controllability of fractional singular dynamicalsystems as treated in the current literature Motivated by thisconsideration in this paper we investigate the reachabilityand controllability of fractional singular dynamical systemswith control delay

In this paper we consider the reachability and controlla-bility of the following fractional singular dynamical systemswith control delay

119864119888

119863120572

119909 (119905) = 119860119909 (119905) + 119861119906 (119905) + 119862119906 (119905 minus 120591) 119905 ge 0

119906 (119905) = 120595 (119905) minus120591 le 119905 le 0

119909 (0) = 1199090

(1)

where 119888119863120572119909(119905) denotes an 120572 order Caputo fractional deriva-tive of 119909(119905) and 0 lt 120572 le 1 119864 119860 119861 and 119862 are the knownconstant matrices 119864119860 isin R119899times119899 119861 isin R119899times119898 119862 isin R119899times119898 andrank(119864) lt 119899119909 isin R119899 is the state variable119906 isin R119898 is the controlinput 120591 gt 0 is the time control delay and 120595(119905) is the initialcontrol function

Themain purpose of this paper is to establish reachabilityand controllability criteria for system (1) based on the alge-braic approach The factors of fractional singular dynamicalsystems including the Caputorsquos fractional derivative controldelay and singular coefficientmatrices are taken into accountsynchronously As discussed by Dai [1] we transform system(1) into two subsystems by the first equivalent form (FE1)which is more convenient to characterize the state reachableset The state response reachable set and sufficient andnecessary conditions for the controllability are obtainedrespectively The proposed criteria are applicable to a largerclass of fractional dynamical systems Therefore we extendthe known results [1 23 25] to a more general case

This paper is organized as follows In the next sectionwe recall some definitions and preliminary facts used inthe paper In Section 3 the state structure of system (1) ischaracterized by analysing the state response and reachableset In Section 4 the necessary and sufficient conditions ofcontrollability for two subsystems and system (1) are derivedrespectively In Section 5 an example is provided to illustratethe effectiveness and applicability of the proposed criteriaFinally some concluding remarks are drawn in Section 6

2 Preliminaries

In this section we first recall some definitions of fractionalcalculus and preliminary facts For more details one can see[9ndash12] Next the first equivalent form (FE1) of system (1) isgiven

(a) The Riemann-Liouvillersquos fractional integral of order120572 gt 0 with the lower limit zero for a function 119891 119877

+

rarr 119877119899

is defined as

119863minus120572

119891 (119905) =1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 gt 0 (2)

provided that the right side is pointwise defined on [0 +infin)where Γ(sdot) is the Gamma function

(b) The Caputorsquos fractional derivative of order 120572 for afunction 119891 119877

+

rarr 119877119899 is defined as

119888

119863120572

119891 (119905) =1

Γ (119898 minus 120572 + 1)int

119905

0

(119905 minus 119904)119898minus120572

119891(119898+1)

(119904) 119889119904

0 le 119898 le 120572 lt 119898 + 1

(3)

For 0 lt 120572 le 1

119863minus120572

119888

119863120572

119891 (119905) = 119891 (119905) minus 119891 (0) (4)

(c) The Mittag-Leffler function in two parameters isdefined as

119864120572120573

(119911) =

+infin

sum

119896=0

119911119896

Γ (119896120572 + 120573) (5)

where 120572 gt 0 120573 gt 0 and 119911 isin C C denotes the complex planeIn particular for 120573 = 1

1198641205721

(119911) = 119864120572(119911) =

+infin

sum

119896=0

119911119896

Γ (119896120572 + 1)(6)

has the interesting property

119888

119863120572

119864120572(120582119911120572

) = 120582119864120572(120582119911120572

) 120582 119911 isin C (7)

(d)The Laplace transform of a function 119891(119905) is defined as

119865 (119904) = pound [119891 (119905)] = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 119904 isin C (8)

where 119891(119905) is 119899-dimensional vector-valued function For 120572 isin

(0 1] it follows from [10] that

pound [(119888119863120572119891) (119905)] = 119904120572pound [119891 (119905)] minus 119904

120572minus1

119891 (0) (9)

(e) For 119899 isin N the sequential fractional derivative forsuitable function 119910(119909) is defined by

119910(119896)

= (D119896120572119910) (119909) = (D120572D(119896minus1)120572119910) (119909) (10)

where 119896 = 1 119899 andD120572 is any fractional differential oper-ator here we still mention it as 119888119863120572

Now we introduce the first equivalent form (FE1) ofsystem (1) by the nonsingular transform which is also calledthe standard decomposition of a singular system

Journal of Applied Mathematics 3

Assume that (119864 119860) is regular throughout this paper From[1 3] there exist two nonsingular matrices 119875119876 isin R119899times119899 suchthat system (1) is equivalent to canonical system as follows

119888

119863120572

1199091(119905) = 119860

11199091(119905) + 119861

1119906 (119905) + 119862

1119906 (119905 minus 120591) 119905 ge 0

119873119888

119863120572

1199092(119905) = 119909

2(119905) + 119861

2119906 (119905) + 119862

2119906 (119905 minus 120591) 119905 ge 0

119906 (119905) = 120595 (119905) minus120591 le 119905 le 0

1199091(0) = 119909

10 119909

2(0) = 119909

20

(11)

where 1199091isin R1198991 119909

2isin R1198992 119899

1+ 1198992= 119899 and 119873 is nilpotent

whose nilpotent index is denoted by ] that is 119873]= 0

119873]minus1

= 0 and

119875119864119876 = [1198681198991

0

0 119873] 119875119860119876 = [

1198601

0

0 1198681198992

]

119875119861 = [1198611

1198612

] 119875119862 = [1198621

1198622

] 119876minus1

119909 (119905) = [1199091(119905)

1199092(119905)

]

(12)

Let ℎ = [120572]] + 1 where [120572]] denotes the integral partof 120572] Let Cℎ

119901([0 +infin)R119898) be the set of ℎ times piecewise

continuously differentiable functions mapping the interval[0 +infin) into R119898 Moreover Cℎ([minus120591 0]R119898) denotes the setof ℎ times continuously differentiable functions mapping theinterval [minus120591 0] into R119898

Applying the method in [3] we can obtain the preciseform of the admissible initial state set 119878(120595) for system (11)For 120595 isin Cℎ([minus120591 0]R119898) we have

119878 (120595)

= 119909 isin R119899

| 119909 = [1199091

1199092

] 1199091isin R1198991

1199092= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

120595 (0) + 119873119896

1198622

119888

119863119896120572

120595 (minus120591)]

(13)

Thus the set of admissible initial data is

A = (1199090 120595) | 120595 isin Cℎ ([minus120591 0] R119898) 119909

0isin 119878 (120595) (14)

3 State Response and State Reachability

In this section we characterize the state structure of system(11) by analysing the state response and reachable set

Theorem 1 For any admissible initial data (1199090 120595) isin A and

the control function 119906(119905) isin Cℎ119901([0 +infin)R119898) the state response

for system (11) can be represented in the following form1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905minus 119904)120572minus1

119864120572120572

[1198601(119905minus 119904)

120572

] 1198611+ (119905minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

119905 ge 0

(15)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] 119905 ge 0

(16)

where119864120572120573

(sdot) is theMittag-Leffler function and 119888119863119896120572119906(sdot) is thesequential fractional derivative

Proof From [10] the Laplace transformof the120572orderCaputofractional derivative of function 119909(119905) is

pound [119863120572119909 (119905)] = 119904120572pound [119909 (119905)] minus 119904

120572minus1

119909 (0) (17)Taking the Laplace transform with respect to 119905 in both sidesof the first equation of system (11) we obtain

119904120572pound [1199091(119905)] minus 119904

120572minus1

1199091(0)

= 1198601pound [1199091(119905)] + 119861

1pound [119906 (119905)] + 119862

1pound [119906 (119905 minus 120591)]

(18)

Hence

pound [1199091(119905)] = (119904

120572

119868 minus 1198601)minus1

119904120572minus1

1199091(0)

+ (119904120572

119868 minus 1198601)minus1pound [119861

1119906 (119905) + 119862

1119906 (119905 minus 120591)]

(19)

From [10] we havepound [119905120573minus1119864

120572120573(1198601119905120572

)] = 119904120572minus120573

(119904120572

119868 minus 1198601)minus1

(20)

Then (19) is equivalent topound [1199091(119905)] = pound [119864

1205721(1198601119905120572

)] 1199091(0)

+ pound [119905120572minus1119864120572120572

(1198601119905120572

)] pound [1198611119906 (119905) + 119862

1119906 (119905 minus 120591)]

(21)The convolution theorem of the Laplace transform applied to(21) yields the form

pound [1199091(119905)] = pound [119864

1205721(1198601119905120572

)] 1199091(0)

+ poundint119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

(22)

4 Journal of Applied Mathematics

Applying the inverse Laplace transform we obtain

1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(23)

On the other hand inserting (16) into both sides of the secondequation in system (11) yields

119873119888

119863120572

1199092(119905) minus 119909

2(119905) minus 119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591)

= minus

]minus1

sum

119896=0

[119873119896+1

1198612

119888

119863(119896+1)120572

119906 (119905) + 119873119896+1

1198622

119888

119863(119896+1)120572

119906 (119905 minus 120591)]

minus 1198612119906 (119905) minus 119862

2119906 (119905 minus 120591)

+

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

= 0

(24)

Then (16) is a solution of the second equation in system (11)Next we will show that any solution of the second

equation in system (11) has the form of (16) Define linearoperator 119879 C1([0 +infin)R1198992) rarr C([0 +infin)R1198992) by1198791199092(119905) =

119888

119863120572

1199092(119905) Then the second equation in system (11)

becomes

(119868 minus 119873119879) 1199092(119905) = minus119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591) (25)

Since 119873 is nilpotent 119873 and 119879 can exchange and (119868 minus

119873119879)(sum]minus1119896=0

119873119896

119879119896

) = 119868 then we have

1199092(119905) = minus(119868 minus 119873119879)

minus1

[1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

+infin

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

]minus1

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

(26)

Thus (16) is indeed a solution of the second equation insystem (11)

Therefore the proof of Theorem 1 is completed

For convenience sake some notations are introduced asfollows

Given matrices 119860 isin R119899times119899 and 119861 119862 isin R119899times119898 denote Im(119861)

as the range of 119861 that is

Im (119861) = 119910 | 119910 = 119861119909 forall119909 isin R119898

(27)

Let

⟨119860 | 119861 119862⟩ = Im (119861) + Im (119860119861) + sdot sdot sdot + Im (119860119899minus1

119861)

+ Im (119862) + Im (119860119862) + sdot sdot sdot + Im (119860119899minus1

119862)

(28)

Then the space ⟨1198601| 1198611 1198621⟩ is spanned by the columns of

[1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] (29)

and the space ⟨119873|1198612 1198622⟩ is spanned by the columns of

[1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] (30)

For any nonzero polynomial 119891(119904) we define119882(119891 119905) R1198991 rarr

R1198991 by

119882(119891 119905) 119911

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus119904)

120572minus1

119864120572120572

[119860lowast

1(119905minus 119904)

120572

] 119891 (119904) 119911 119889119904

(31)

where lowast denotes the matrix transposeThe following lemma is required for the main result

which is a natural extension of Lemma 2-11 in [1]

Lemma 2 Given matrices 1198601isin R1198991times1198991 and 119861

1 1198621isin R1198991times1198981

then

Im119882(119891 119905) = ⟨1198601| 1198611 1198621⟩ (32)

Proof The formula (32) holds if and only if

Ker119882(119891 119905) =

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(33)

Journal of Applied Mathematics 5

If 1199110isin Ker119882(119891 119905) and 119911

0= 0 119911lowast0119866(119905)1199110= 0 that is

int

119905

119905minus120591

10038171003817100381710038171003817119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[(119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904

+ int

119905minus120591

0

10038171003817100381710038171003817[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904 = 0

(34)

then we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0= 0 119904 isin [119905 minus 120591 119905]

(35)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119891 (119904) 119911

0= 0 119904 isin [0 119905 minus 120591]

(36)

Since polynomial 119891(119904) only has a finite number of zero on119905 minus 120591 le 119904 le 119905 and 0 le 119904 le 119905 minus 120591 we immediately have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119911

0= 0 119904 isin [119905 minus 120591 119905] (37)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0= 0 119904 isin [0 119905 minus 120591]

(38)

Repeatedly taking the Caputorsquos derivative on both sides of(37) we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] (119860

lowast

1)119894

1199110= 0

119894 = 1 2 1198991minus 1 119904 isin [119905 minus 120591 119905]

(39)

Substituting 119904 = 119905 in (39) yields

119861lowast

(119860lowast

)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (40)

which implies that 1199110isin Ker119861lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

According to Cayley-Hamilton theorem [4] (119905 minus

119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] can be represented as

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] =

+infin

sum

119896=0

(119905 minus 119904)119896120572+120572minus1

Γ (119896120572 + 120572)119860119896

1

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119860

119896

1

(41)

For 119904 isin [0 119905 minus 120591] it follows from (39) and (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(42)

Inserting (42) into (38) yields

119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) 119911

0= 0 (43)

Repeatedly taking the Caputorsquos fractional derivative on bothsides of (43) and letting 119904 = 119905 minus 120591 we have

119862lowast

1119860lowast

1

119896

1199110= 0 119896 = 1 2 119899

1minus 1 (44)

which implies that 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

Therefore we have

Ker119882(119891 119905) sube

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(45)

Conversely if 1199110isin ⋂1198991minus1

119894=0Ker119861lowast1(119860lowast

1)119894

⋂1198991minus1

119894=0Ker119862lowast

1(119860lowast

1)119894

and 1199110

= 0 then 1199110isin Ker119861lowast

1(119860lowast

1)119894 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 =

1 2 1198991minus 1 and

119861lowast

1(119860lowast

1)119894

1199110= 119862lowast

1(119860lowast

1)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (46)

For 119904 isin [119905 minus 120591 119905] it follows from (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(47)

For 119904 isin [0 119905 minus 120591] the same argument yields

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110

+

1198991minus1

sum

119896=0

120582119896(119905 minus 120591 minus 119904) 119862

lowast

1119860lowast

1

119896

1199110= 0

(48)

Hence 1199110isin Ker119882(119891 119905) which implies

Ker119882(119891 119905) supe

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(49)

From (45) and (49) we know that (32) is true The proof istherefore completed

The reachable set for system (1) (or (11)) may be definedas follows

Definition 3 Any vector 120596 isin R119899 in 119899-dimensional vectorspace is said to reachable if there exists an admissibleinitial data (119909

0 120595) isin A admissible control input 119906(119905) isin

Cℎ119901([0 +infin)R119898) and 119905

119891gt 0 such that the solution of system

(1) (or (11)) satisfies 119909(119905119891 1199090 120595) = 120596

Let R(1199090 120595) be the reachable set from any admissible

initial data (1199090 120595) isin A Then

R (1199090 120595)

= 120596 isin R119899

| (1199090 120595) isin A 119906 (119905) isin Cℎ

119901([0 +infin) R

119898

)

119905119891gt 0 119909 (119905

119891 1199090 120595) = 120596

(50)

6 Journal of Applied Mathematics

Considering the reachable set from the initial conditions1199090= 0 and 120595 equiv 0 we have the following theorem

Theorem 4 For system (11) the reachable set R(0 0) can berepresented as

R (0 0) = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (51)

where ldquooplusrdquo is the direct sum in vector space

Proof Let 120596 isin R(0 0) from (15) (16) and (50) then thereexists 119905

119891gt 0 and 119906(119905) isin Cℎ

119901([0 +infin)R119898) such that

1205961= int

119905119891

0

(119905119891minus 119904)120572minus1

119864120572120572

[1198601(119905119891minus 119904)120572

] 1198611119906 (119904) 119889119904

+ int

119905119891minus120591

0

(119905119891minus 120591 minus 119904)

120572minus1

119864120572120572

[1198601(119905119891minus 120591 minus 119904)

120572

]

times 1198621119906 (119904) 119889119904

(52)

Combining (41) and (52) yields

1205961=

1198991minus1

sum

119896=0

int

119905119891

0

120593119896(119905119891minus 119904)119860

119896

11198611119906 (119904) 119889119904

+

1198991minus1

sum

119896=0

int

119905119891minus120591

0

120601119896(119905119891minus 120591 minus 119904)119860

119896

11198621119906 (119904) 119889119904

(53)

Therefore 1205961isin ⟨1198601| 1198611 1198621⟩ Moreover we also have

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905119891) + 119873

119896

1198622

119888

119863119896120572

119906 (119905119891minus 120591)]

(54)

which implies that 1205962isin ⟨119873 | 119861

2 1198622⟩ Thus

R (0 0) sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (55)

On the other hand we need to prove

R (0 0) supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (56)

Suppose that

120596 = [1205961

1205962

] isin ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (57)

where 1205961isin ⟨1198601| 1198611 1198621⟩ 1205962isin ⟨119873 | 119861

2 1198622⟩ and 120596

1= 0

1205962

= 0 For the initial conditions 1199090= 0 and 120595(119905) equiv 0 119905 isin

[minus120591 0] let 119906(119904) = 1199061(119904) + 119906

2(119904) then we have

1199091(119905) = int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 1198621[1199061(119904) + 119906

2(119904)] 119889119904

(58)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199061(119905) + 119873

119896

1198622

119888

119863119896120572

1199061(119905 minus 120591)]

minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199062(119905) + 119873

119896

1198622

119888

119863119896120572

1199062(119905 minus 120591)]

(59)

As discussed by Yip and Sincovec [3] we choose 1199061(119904) =

119891(119904)119910(119904) to satisfy

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119891 (119904) 119892 (119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198621119891 (119904 minus 120591) 119892 (119904 minus 120591) 119889119904

= 1205961minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199062(119904) 119889119904

minus int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 11986211199062(119904) 119889119904 equiv

1

119888

119863119896120572

1199061(119905) =

119888

119863119896120572

1199061(119905 minus 120591) = 0 119896 = 0 1 2 120592 minus 1

(60)

Then 1199061(119904) contributes nothing to 119909

2(119905) at 119904 = 119905 and 119904 = 119905 minus 120591

Now we prove that (60) is trueIn fact for

1isin ⟨1198601| 1198611 1198621⟩ fromLemma 2 there exists

119911 isin R119899 such that119882(119891 119905)119911 = 1

Let

119910 (119904) =

119891 (119904) [119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904))

+119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0 le 119904 le 119905 minus 120591

119891 (119904) 119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

119905 minus 120591 le 119904 le 119905 minus120591 le 119904 le 0

(61)

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom 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

2 Journal of Applied Mathematics

developed the control theory of fractional dynamical systemsThe earlier studies concerning the controllability of fractionaldynamical systems with control delay can be found in [25 2729] which especially focused on fractional normal systemsHowever it should be emphasized that the control theory offractional singular systems is not yet sufficiently elaboratedcompared to that of fractional normal systems In this regardit is necessary and important to study the controllabilityproblems for fractional singular dynamical systems To thebest of our knowledge there are no relevant reports on reach-ability and controllability of fractional singular dynamicalsystems as treated in the current literature Motivated by thisconsideration in this paper we investigate the reachabilityand controllability of fractional singular dynamical systemswith control delay

In this paper we consider the reachability and controlla-bility of the following fractional singular dynamical systemswith control delay

119864119888

119863120572

119909 (119905) = 119860119909 (119905) + 119861119906 (119905) + 119862119906 (119905 minus 120591) 119905 ge 0

119906 (119905) = 120595 (119905) minus120591 le 119905 le 0

119909 (0) = 1199090

(1)

where 119888119863120572119909(119905) denotes an 120572 order Caputo fractional deriva-tive of 119909(119905) and 0 lt 120572 le 1 119864 119860 119861 and 119862 are the knownconstant matrices 119864119860 isin R119899times119899 119861 isin R119899times119898 119862 isin R119899times119898 andrank(119864) lt 119899119909 isin R119899 is the state variable119906 isin R119898 is the controlinput 120591 gt 0 is the time control delay and 120595(119905) is the initialcontrol function

Themain purpose of this paper is to establish reachabilityand controllability criteria for system (1) based on the alge-braic approach The factors of fractional singular dynamicalsystems including the Caputorsquos fractional derivative controldelay and singular coefficientmatrices are taken into accountsynchronously As discussed by Dai [1] we transform system(1) into two subsystems by the first equivalent form (FE1)which is more convenient to characterize the state reachableset The state response reachable set and sufficient andnecessary conditions for the controllability are obtainedrespectively The proposed criteria are applicable to a largerclass of fractional dynamical systems Therefore we extendthe known results [1 23 25] to a more general case

This paper is organized as follows In the next sectionwe recall some definitions and preliminary facts used inthe paper In Section 3 the state structure of system (1) ischaracterized by analysing the state response and reachableset In Section 4 the necessary and sufficient conditions ofcontrollability for two subsystems and system (1) are derivedrespectively In Section 5 an example is provided to illustratethe effectiveness and applicability of the proposed criteriaFinally some concluding remarks are drawn in Section 6

2 Preliminaries

In this section we first recall some definitions of fractionalcalculus and preliminary facts For more details one can see[9ndash12] Next the first equivalent form (FE1) of system (1) isgiven

(a) The Riemann-Liouvillersquos fractional integral of order120572 gt 0 with the lower limit zero for a function 119891 119877

+

rarr 119877119899

is defined as

119863minus120572

119891 (119905) =1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 gt 0 (2)

provided that the right side is pointwise defined on [0 +infin)where Γ(sdot) is the Gamma function

(b) The Caputorsquos fractional derivative of order 120572 for afunction 119891 119877

+

rarr 119877119899 is defined as

119888

119863120572

119891 (119905) =1

Γ (119898 minus 120572 + 1)int

119905

0

(119905 minus 119904)119898minus120572

119891(119898+1)

(119904) 119889119904

0 le 119898 le 120572 lt 119898 + 1

(3)

For 0 lt 120572 le 1

119863minus120572

119888

119863120572

119891 (119905) = 119891 (119905) minus 119891 (0) (4)

(c) The Mittag-Leffler function in two parameters isdefined as

119864120572120573

(119911) =

+infin

sum

119896=0

119911119896

Γ (119896120572 + 120573) (5)

where 120572 gt 0 120573 gt 0 and 119911 isin C C denotes the complex planeIn particular for 120573 = 1

1198641205721

(119911) = 119864120572(119911) =

+infin

sum

119896=0

119911119896

Γ (119896120572 + 1)(6)

has the interesting property

119888

119863120572

119864120572(120582119911120572

) = 120582119864120572(120582119911120572

) 120582 119911 isin C (7)

(d)The Laplace transform of a function 119891(119905) is defined as

119865 (119904) = pound [119891 (119905)] = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 119904 isin C (8)

where 119891(119905) is 119899-dimensional vector-valued function For 120572 isin

(0 1] it follows from [10] that

pound [(119888119863120572119891) (119905)] = 119904120572pound [119891 (119905)] minus 119904

120572minus1

119891 (0) (9)

(e) For 119899 isin N the sequential fractional derivative forsuitable function 119910(119909) is defined by

119910(119896)

= (D119896120572119910) (119909) = (D120572D(119896minus1)120572119910) (119909) (10)

where 119896 = 1 119899 andD120572 is any fractional differential oper-ator here we still mention it as 119888119863120572

Now we introduce the first equivalent form (FE1) ofsystem (1) by the nonsingular transform which is also calledthe standard decomposition of a singular system

Journal of Applied Mathematics 3

Assume that (119864 119860) is regular throughout this paper From[1 3] there exist two nonsingular matrices 119875119876 isin R119899times119899 suchthat system (1) is equivalent to canonical system as follows

119888

119863120572

1199091(119905) = 119860

11199091(119905) + 119861

1119906 (119905) + 119862

1119906 (119905 minus 120591) 119905 ge 0

119873119888

119863120572

1199092(119905) = 119909

2(119905) + 119861

2119906 (119905) + 119862

2119906 (119905 minus 120591) 119905 ge 0

119906 (119905) = 120595 (119905) minus120591 le 119905 le 0

1199091(0) = 119909

10 119909

2(0) = 119909

20

(11)

where 1199091isin R1198991 119909

2isin R1198992 119899

1+ 1198992= 119899 and 119873 is nilpotent

whose nilpotent index is denoted by ] that is 119873]= 0

119873]minus1

= 0 and

119875119864119876 = [1198681198991

0

0 119873] 119875119860119876 = [

1198601

0

0 1198681198992

]

119875119861 = [1198611

1198612

] 119875119862 = [1198621

1198622

] 119876minus1

119909 (119905) = [1199091(119905)

1199092(119905)

]

(12)

Let ℎ = [120572]] + 1 where [120572]] denotes the integral partof 120572] Let Cℎ

119901([0 +infin)R119898) be the set of ℎ times piecewise

continuously differentiable functions mapping the interval[0 +infin) into R119898 Moreover Cℎ([minus120591 0]R119898) denotes the setof ℎ times continuously differentiable functions mapping theinterval [minus120591 0] into R119898

Applying the method in [3] we can obtain the preciseform of the admissible initial state set 119878(120595) for system (11)For 120595 isin Cℎ([minus120591 0]R119898) we have

119878 (120595)

= 119909 isin R119899

| 119909 = [1199091

1199092

] 1199091isin R1198991

1199092= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

120595 (0) + 119873119896

1198622

119888

119863119896120572

120595 (minus120591)]

(13)

Thus the set of admissible initial data is

A = (1199090 120595) | 120595 isin Cℎ ([minus120591 0] R119898) 119909

0isin 119878 (120595) (14)

3 State Response and State Reachability

In this section we characterize the state structure of system(11) by analysing the state response and reachable set

Theorem 1 For any admissible initial data (1199090 120595) isin A and

the control function 119906(119905) isin Cℎ119901([0 +infin)R119898) the state response

for system (11) can be represented in the following form1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905minus 119904)120572minus1

119864120572120572

[1198601(119905minus 119904)

120572

] 1198611+ (119905minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

119905 ge 0

(15)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] 119905 ge 0

(16)

where119864120572120573

(sdot) is theMittag-Leffler function and 119888119863119896120572119906(sdot) is thesequential fractional derivative

Proof From [10] the Laplace transformof the120572orderCaputofractional derivative of function 119909(119905) is

pound [119863120572119909 (119905)] = 119904120572pound [119909 (119905)] minus 119904

120572minus1

119909 (0) (17)Taking the Laplace transform with respect to 119905 in both sidesof the first equation of system (11) we obtain

119904120572pound [1199091(119905)] minus 119904

120572minus1

1199091(0)

= 1198601pound [1199091(119905)] + 119861

1pound [119906 (119905)] + 119862

1pound [119906 (119905 minus 120591)]

(18)

Hence

pound [1199091(119905)] = (119904

120572

119868 minus 1198601)minus1

119904120572minus1

1199091(0)

+ (119904120572

119868 minus 1198601)minus1pound [119861

1119906 (119905) + 119862

1119906 (119905 minus 120591)]

(19)

From [10] we havepound [119905120573minus1119864

120572120573(1198601119905120572

)] = 119904120572minus120573

(119904120572

119868 minus 1198601)minus1

(20)

Then (19) is equivalent topound [1199091(119905)] = pound [119864

1205721(1198601119905120572

)] 1199091(0)

+ pound [119905120572minus1119864120572120572

(1198601119905120572

)] pound [1198611119906 (119905) + 119862

1119906 (119905 minus 120591)]

(21)The convolution theorem of the Laplace transform applied to(21) yields the form

pound [1199091(119905)] = pound [119864

1205721(1198601119905120572

)] 1199091(0)

+ poundint119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

(22)

4 Journal of Applied Mathematics

Applying the inverse Laplace transform we obtain

1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(23)

On the other hand inserting (16) into both sides of the secondequation in system (11) yields

119873119888

119863120572

1199092(119905) minus 119909

2(119905) minus 119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591)

= minus

]minus1

sum

119896=0

[119873119896+1

1198612

119888

119863(119896+1)120572

119906 (119905) + 119873119896+1

1198622

119888

119863(119896+1)120572

119906 (119905 minus 120591)]

minus 1198612119906 (119905) minus 119862

2119906 (119905 minus 120591)

+

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

= 0

(24)

Then (16) is a solution of the second equation in system (11)Next we will show that any solution of the second

equation in system (11) has the form of (16) Define linearoperator 119879 C1([0 +infin)R1198992) rarr C([0 +infin)R1198992) by1198791199092(119905) =

119888

119863120572

1199092(119905) Then the second equation in system (11)

becomes

(119868 minus 119873119879) 1199092(119905) = minus119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591) (25)

Since 119873 is nilpotent 119873 and 119879 can exchange and (119868 minus

119873119879)(sum]minus1119896=0

119873119896

119879119896

) = 119868 then we have

1199092(119905) = minus(119868 minus 119873119879)

minus1

[1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

+infin

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

]minus1

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

(26)

Thus (16) is indeed a solution of the second equation insystem (11)

Therefore the proof of Theorem 1 is completed

For convenience sake some notations are introduced asfollows

Given matrices 119860 isin R119899times119899 and 119861 119862 isin R119899times119898 denote Im(119861)

as the range of 119861 that is

Im (119861) = 119910 | 119910 = 119861119909 forall119909 isin R119898

(27)

Let

⟨119860 | 119861 119862⟩ = Im (119861) + Im (119860119861) + sdot sdot sdot + Im (119860119899minus1

119861)

+ Im (119862) + Im (119860119862) + sdot sdot sdot + Im (119860119899minus1

119862)

(28)

Then the space ⟨1198601| 1198611 1198621⟩ is spanned by the columns of

[1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] (29)

and the space ⟨119873|1198612 1198622⟩ is spanned by the columns of

[1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] (30)

For any nonzero polynomial 119891(119904) we define119882(119891 119905) R1198991 rarr

R1198991 by

119882(119891 119905) 119911

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus119904)

120572minus1

119864120572120572

[119860lowast

1(119905minus 119904)

120572

] 119891 (119904) 119911 119889119904

(31)

where lowast denotes the matrix transposeThe following lemma is required for the main result

which is a natural extension of Lemma 2-11 in [1]

Lemma 2 Given matrices 1198601isin R1198991times1198991 and 119861

1 1198621isin R1198991times1198981

then

Im119882(119891 119905) = ⟨1198601| 1198611 1198621⟩ (32)

Proof The formula (32) holds if and only if

Ker119882(119891 119905) =

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(33)

Journal of Applied Mathematics 5

If 1199110isin Ker119882(119891 119905) and 119911

0= 0 119911lowast0119866(119905)1199110= 0 that is

int

119905

119905minus120591

10038171003817100381710038171003817119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[(119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904

+ int

119905minus120591

0

10038171003817100381710038171003817[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904 = 0

(34)

then we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0= 0 119904 isin [119905 minus 120591 119905]

(35)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119891 (119904) 119911

0= 0 119904 isin [0 119905 minus 120591]

(36)

Since polynomial 119891(119904) only has a finite number of zero on119905 minus 120591 le 119904 le 119905 and 0 le 119904 le 119905 minus 120591 we immediately have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119911

0= 0 119904 isin [119905 minus 120591 119905] (37)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0= 0 119904 isin [0 119905 minus 120591]

(38)

Repeatedly taking the Caputorsquos derivative on both sides of(37) we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] (119860

lowast

1)119894

1199110= 0

119894 = 1 2 1198991minus 1 119904 isin [119905 minus 120591 119905]

(39)

Substituting 119904 = 119905 in (39) yields

119861lowast

(119860lowast

)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (40)

which implies that 1199110isin Ker119861lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

According to Cayley-Hamilton theorem [4] (119905 minus

119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] can be represented as

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] =

+infin

sum

119896=0

(119905 minus 119904)119896120572+120572minus1

Γ (119896120572 + 120572)119860119896

1

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119860

119896

1

(41)

For 119904 isin [0 119905 minus 120591] it follows from (39) and (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(42)

Inserting (42) into (38) yields

119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) 119911

0= 0 (43)

Repeatedly taking the Caputorsquos fractional derivative on bothsides of (43) and letting 119904 = 119905 minus 120591 we have

119862lowast

1119860lowast

1

119896

1199110= 0 119896 = 1 2 119899

1minus 1 (44)

which implies that 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

Therefore we have

Ker119882(119891 119905) sube

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(45)

Conversely if 1199110isin ⋂1198991minus1

119894=0Ker119861lowast1(119860lowast

1)119894

⋂1198991minus1

119894=0Ker119862lowast

1(119860lowast

1)119894

and 1199110

= 0 then 1199110isin Ker119861lowast

1(119860lowast

1)119894 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 =

1 2 1198991minus 1 and

119861lowast

1(119860lowast

1)119894

1199110= 119862lowast

1(119860lowast

1)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (46)

For 119904 isin [119905 minus 120591 119905] it follows from (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(47)

For 119904 isin [0 119905 minus 120591] the same argument yields

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110

+

1198991minus1

sum

119896=0

120582119896(119905 minus 120591 minus 119904) 119862

lowast

1119860lowast

1

119896

1199110= 0

(48)

Hence 1199110isin Ker119882(119891 119905) which implies

Ker119882(119891 119905) supe

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(49)

From (45) and (49) we know that (32) is true The proof istherefore completed

The reachable set for system (1) (or (11)) may be definedas follows

Definition 3 Any vector 120596 isin R119899 in 119899-dimensional vectorspace is said to reachable if there exists an admissibleinitial data (119909

0 120595) isin A admissible control input 119906(119905) isin

Cℎ119901([0 +infin)R119898) and 119905

119891gt 0 such that the solution of system

(1) (or (11)) satisfies 119909(119905119891 1199090 120595) = 120596

Let R(1199090 120595) be the reachable set from any admissible

initial data (1199090 120595) isin A Then

R (1199090 120595)

= 120596 isin R119899

| (1199090 120595) isin A 119906 (119905) isin Cℎ

119901([0 +infin) R

119898

)

119905119891gt 0 119909 (119905

119891 1199090 120595) = 120596

(50)

6 Journal of Applied Mathematics

Considering the reachable set from the initial conditions1199090= 0 and 120595 equiv 0 we have the following theorem

Theorem 4 For system (11) the reachable set R(0 0) can berepresented as

R (0 0) = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (51)

where ldquooplusrdquo is the direct sum in vector space

Proof Let 120596 isin R(0 0) from (15) (16) and (50) then thereexists 119905

119891gt 0 and 119906(119905) isin Cℎ

119901([0 +infin)R119898) such that

1205961= int

119905119891

0

(119905119891minus 119904)120572minus1

119864120572120572

[1198601(119905119891minus 119904)120572

] 1198611119906 (119904) 119889119904

+ int

119905119891minus120591

0

(119905119891minus 120591 minus 119904)

120572minus1

119864120572120572

[1198601(119905119891minus 120591 minus 119904)

120572

]

times 1198621119906 (119904) 119889119904

(52)

Combining (41) and (52) yields

1205961=

1198991minus1

sum

119896=0

int

119905119891

0

120593119896(119905119891minus 119904)119860

119896

11198611119906 (119904) 119889119904

+

1198991minus1

sum

119896=0

int

119905119891minus120591

0

120601119896(119905119891minus 120591 minus 119904)119860

119896

11198621119906 (119904) 119889119904

(53)

Therefore 1205961isin ⟨1198601| 1198611 1198621⟩ Moreover we also have

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905119891) + 119873

119896

1198622

119888

119863119896120572

119906 (119905119891minus 120591)]

(54)

which implies that 1205962isin ⟨119873 | 119861

2 1198622⟩ Thus

R (0 0) sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (55)

On the other hand we need to prove

R (0 0) supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (56)

Suppose that

120596 = [1205961

1205962

] isin ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (57)

where 1205961isin ⟨1198601| 1198611 1198621⟩ 1205962isin ⟨119873 | 119861

2 1198622⟩ and 120596

1= 0

1205962

= 0 For the initial conditions 1199090= 0 and 120595(119905) equiv 0 119905 isin

[minus120591 0] let 119906(119904) = 1199061(119904) + 119906

2(119904) then we have

1199091(119905) = int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 1198621[1199061(119904) + 119906

2(119904)] 119889119904

(58)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199061(119905) + 119873

119896

1198622

119888

119863119896120572

1199061(119905 minus 120591)]

minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199062(119905) + 119873

119896

1198622

119888

119863119896120572

1199062(119905 minus 120591)]

(59)

As discussed by Yip and Sincovec [3] we choose 1199061(119904) =

119891(119904)119910(119904) to satisfy

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119891 (119904) 119892 (119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198621119891 (119904 minus 120591) 119892 (119904 minus 120591) 119889119904

= 1205961minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199062(119904) 119889119904

minus int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 11986211199062(119904) 119889119904 equiv

1

119888

119863119896120572

1199061(119905) =

119888

119863119896120572

1199061(119905 minus 120591) = 0 119896 = 0 1 2 120592 minus 1

(60)

Then 1199061(119904) contributes nothing to 119909

2(119905) at 119904 = 119905 and 119904 = 119905 minus 120591

Now we prove that (60) is trueIn fact for

1isin ⟨1198601| 1198611 1198621⟩ fromLemma 2 there exists

119911 isin R119899 such that119882(119891 119905)119911 = 1

Let

119910 (119904) =

119891 (119904) [119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904))

+119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0 le 119904 le 119905 minus 120591

119891 (119904) 119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

119905 minus 120591 le 119904 le 119905 minus120591 le 119904 le 0

(61)

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Journal of Applied Mathematics 3

Assume that (119864 119860) is regular throughout this paper From[1 3] there exist two nonsingular matrices 119875119876 isin R119899times119899 suchthat system (1) is equivalent to canonical system as follows

119888

119863120572

1199091(119905) = 119860

11199091(119905) + 119861

1119906 (119905) + 119862

1119906 (119905 minus 120591) 119905 ge 0

119873119888

119863120572

1199092(119905) = 119909

2(119905) + 119861

2119906 (119905) + 119862

2119906 (119905 minus 120591) 119905 ge 0

119906 (119905) = 120595 (119905) minus120591 le 119905 le 0

1199091(0) = 119909

10 119909

2(0) = 119909

20

(11)

where 1199091isin R1198991 119909

2isin R1198992 119899

1+ 1198992= 119899 and 119873 is nilpotent

whose nilpotent index is denoted by ] that is 119873]= 0

119873]minus1

= 0 and

119875119864119876 = [1198681198991

0

0 119873] 119875119860119876 = [

1198601

0

0 1198681198992

]

119875119861 = [1198611

1198612

] 119875119862 = [1198621

1198622

] 119876minus1

119909 (119905) = [1199091(119905)

1199092(119905)

]

(12)

Let ℎ = [120572]] + 1 where [120572]] denotes the integral partof 120572] Let Cℎ

119901([0 +infin)R119898) be the set of ℎ times piecewise

continuously differentiable functions mapping the interval[0 +infin) into R119898 Moreover Cℎ([minus120591 0]R119898) denotes the setof ℎ times continuously differentiable functions mapping theinterval [minus120591 0] into R119898

Applying the method in [3] we can obtain the preciseform of the admissible initial state set 119878(120595) for system (11)For 120595 isin Cℎ([minus120591 0]R119898) we have

119878 (120595)

= 119909 isin R119899

| 119909 = [1199091

1199092

] 1199091isin R1198991

1199092= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

120595 (0) + 119873119896

1198622

119888

119863119896120572

120595 (minus120591)]

(13)

Thus the set of admissible initial data is

A = (1199090 120595) | 120595 isin Cℎ ([minus120591 0] R119898) 119909

0isin 119878 (120595) (14)

3 State Response and State Reachability

In this section we characterize the state structure of system(11) by analysing the state response and reachable set

Theorem 1 For any admissible initial data (1199090 120595) isin A and

the control function 119906(119905) isin Cℎ119901([0 +infin)R119898) the state response

for system (11) can be represented in the following form1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905minus 119904)120572minus1

119864120572120572

[1198601(119905minus 119904)

120572

] 1198611+ (119905minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

119905 ge 0

(15)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] 119905 ge 0

(16)

where119864120572120573

(sdot) is theMittag-Leffler function and 119888119863119896120572119906(sdot) is thesequential fractional derivative

Proof From [10] the Laplace transformof the120572orderCaputofractional derivative of function 119909(119905) is

pound [119863120572119909 (119905)] = 119904120572pound [119909 (119905)] minus 119904

120572minus1

119909 (0) (17)Taking the Laplace transform with respect to 119905 in both sidesof the first equation of system (11) we obtain

119904120572pound [1199091(119905)] minus 119904

120572minus1

1199091(0)

= 1198601pound [1199091(119905)] + 119861

1pound [119906 (119905)] + 119862

1pound [119906 (119905 minus 120591)]

(18)

Hence

pound [1199091(119905)] = (119904

120572

119868 minus 1198601)minus1

119904120572minus1

1199091(0)

+ (119904120572

119868 minus 1198601)minus1pound [119861

1119906 (119905) + 119862

1119906 (119905 minus 120591)]

(19)

From [10] we havepound [119905120573minus1119864

120572120573(1198601119905120572

)] = 119904120572minus120573

(119904120572

119868 minus 1198601)minus1

(20)

Then (19) is equivalent topound [1199091(119905)] = pound [119864

1205721(1198601119905120572

)] 1199091(0)

+ pound [119905120572minus1119864120572120572

(1198601119905120572

)] pound [1198611119906 (119905) + 119862

1119906 (119905 minus 120591)]

(21)The convolution theorem of the Laplace transform applied to(21) yields the form

pound [1199091(119905)] = pound [119864

1205721(1198601119905120572

)] 1199091(0)

+ poundint119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

(22)

4 Journal of Applied Mathematics

Applying the inverse Laplace transform we obtain

1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(23)

On the other hand inserting (16) into both sides of the secondequation in system (11) yields

119873119888

119863120572

1199092(119905) minus 119909

2(119905) minus 119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591)

= minus

]minus1

sum

119896=0

[119873119896+1

1198612

119888

119863(119896+1)120572

119906 (119905) + 119873119896+1

1198622

119888

119863(119896+1)120572

119906 (119905 minus 120591)]

minus 1198612119906 (119905) minus 119862

2119906 (119905 minus 120591)

+

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

= 0

(24)

Then (16) is a solution of the second equation in system (11)Next we will show that any solution of the second

equation in system (11) has the form of (16) Define linearoperator 119879 C1([0 +infin)R1198992) rarr C([0 +infin)R1198992) by1198791199092(119905) =

119888

119863120572

1199092(119905) Then the second equation in system (11)

becomes

(119868 minus 119873119879) 1199092(119905) = minus119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591) (25)

Since 119873 is nilpotent 119873 and 119879 can exchange and (119868 minus

119873119879)(sum]minus1119896=0

119873119896

119879119896

) = 119868 then we have

1199092(119905) = minus(119868 minus 119873119879)

minus1

[1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

+infin

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

]minus1

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

(26)

Thus (16) is indeed a solution of the second equation insystem (11)

Therefore the proof of Theorem 1 is completed

For convenience sake some notations are introduced asfollows

Given matrices 119860 isin R119899times119899 and 119861 119862 isin R119899times119898 denote Im(119861)

as the range of 119861 that is

Im (119861) = 119910 | 119910 = 119861119909 forall119909 isin R119898

(27)

Let

⟨119860 | 119861 119862⟩ = Im (119861) + Im (119860119861) + sdot sdot sdot + Im (119860119899minus1

119861)

+ Im (119862) + Im (119860119862) + sdot sdot sdot + Im (119860119899minus1

119862)

(28)

Then the space ⟨1198601| 1198611 1198621⟩ is spanned by the columns of

[1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] (29)

and the space ⟨119873|1198612 1198622⟩ is spanned by the columns of

[1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] (30)

For any nonzero polynomial 119891(119904) we define119882(119891 119905) R1198991 rarr

R1198991 by

119882(119891 119905) 119911

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus119904)

120572minus1

119864120572120572

[119860lowast

1(119905minus 119904)

120572

] 119891 (119904) 119911 119889119904

(31)

where lowast denotes the matrix transposeThe following lemma is required for the main result

which is a natural extension of Lemma 2-11 in [1]

Lemma 2 Given matrices 1198601isin R1198991times1198991 and 119861

1 1198621isin R1198991times1198981

then

Im119882(119891 119905) = ⟨1198601| 1198611 1198621⟩ (32)

Proof The formula (32) holds if and only if

Ker119882(119891 119905) =

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(33)

Journal of Applied Mathematics 5

If 1199110isin Ker119882(119891 119905) and 119911

0= 0 119911lowast0119866(119905)1199110= 0 that is

int

119905

119905minus120591

10038171003817100381710038171003817119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[(119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904

+ int

119905minus120591

0

10038171003817100381710038171003817[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904 = 0

(34)

then we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0= 0 119904 isin [119905 minus 120591 119905]

(35)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119891 (119904) 119911

0= 0 119904 isin [0 119905 minus 120591]

(36)

Since polynomial 119891(119904) only has a finite number of zero on119905 minus 120591 le 119904 le 119905 and 0 le 119904 le 119905 minus 120591 we immediately have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119911

0= 0 119904 isin [119905 minus 120591 119905] (37)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0= 0 119904 isin [0 119905 minus 120591]

(38)

Repeatedly taking the Caputorsquos derivative on both sides of(37) we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] (119860

lowast

1)119894

1199110= 0

119894 = 1 2 1198991minus 1 119904 isin [119905 minus 120591 119905]

(39)

Substituting 119904 = 119905 in (39) yields

119861lowast

(119860lowast

)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (40)

which implies that 1199110isin Ker119861lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

According to Cayley-Hamilton theorem [4] (119905 minus

119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] can be represented as

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] =

+infin

sum

119896=0

(119905 minus 119904)119896120572+120572minus1

Γ (119896120572 + 120572)119860119896

1

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119860

119896

1

(41)

For 119904 isin [0 119905 minus 120591] it follows from (39) and (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(42)

Inserting (42) into (38) yields

119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) 119911

0= 0 (43)

Repeatedly taking the Caputorsquos fractional derivative on bothsides of (43) and letting 119904 = 119905 minus 120591 we have

119862lowast

1119860lowast

1

119896

1199110= 0 119896 = 1 2 119899

1minus 1 (44)

which implies that 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

Therefore we have

Ker119882(119891 119905) sube

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(45)

Conversely if 1199110isin ⋂1198991minus1

119894=0Ker119861lowast1(119860lowast

1)119894

⋂1198991minus1

119894=0Ker119862lowast

1(119860lowast

1)119894

and 1199110

= 0 then 1199110isin Ker119861lowast

1(119860lowast

1)119894 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 =

1 2 1198991minus 1 and

119861lowast

1(119860lowast

1)119894

1199110= 119862lowast

1(119860lowast

1)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (46)

For 119904 isin [119905 minus 120591 119905] it follows from (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(47)

For 119904 isin [0 119905 minus 120591] the same argument yields

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110

+

1198991minus1

sum

119896=0

120582119896(119905 minus 120591 minus 119904) 119862

lowast

1119860lowast

1

119896

1199110= 0

(48)

Hence 1199110isin Ker119882(119891 119905) which implies

Ker119882(119891 119905) supe

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(49)

From (45) and (49) we know that (32) is true The proof istherefore completed

The reachable set for system (1) (or (11)) may be definedas follows

Definition 3 Any vector 120596 isin R119899 in 119899-dimensional vectorspace is said to reachable if there exists an admissibleinitial data (119909

0 120595) isin A admissible control input 119906(119905) isin

Cℎ119901([0 +infin)R119898) and 119905

119891gt 0 such that the solution of system

(1) (or (11)) satisfies 119909(119905119891 1199090 120595) = 120596

Let R(1199090 120595) be the reachable set from any admissible

initial data (1199090 120595) isin A Then

R (1199090 120595)

= 120596 isin R119899

| (1199090 120595) isin A 119906 (119905) isin Cℎ

119901([0 +infin) R

119898

)

119905119891gt 0 119909 (119905

119891 1199090 120595) = 120596

(50)

6 Journal of Applied Mathematics

Considering the reachable set from the initial conditions1199090= 0 and 120595 equiv 0 we have the following theorem

Theorem 4 For system (11) the reachable set R(0 0) can berepresented as

R (0 0) = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (51)

where ldquooplusrdquo is the direct sum in vector space

Proof Let 120596 isin R(0 0) from (15) (16) and (50) then thereexists 119905

119891gt 0 and 119906(119905) isin Cℎ

119901([0 +infin)R119898) such that

1205961= int

119905119891

0

(119905119891minus 119904)120572minus1

119864120572120572

[1198601(119905119891minus 119904)120572

] 1198611119906 (119904) 119889119904

+ int

119905119891minus120591

0

(119905119891minus 120591 minus 119904)

120572minus1

119864120572120572

[1198601(119905119891minus 120591 minus 119904)

120572

]

times 1198621119906 (119904) 119889119904

(52)

Combining (41) and (52) yields

1205961=

1198991minus1

sum

119896=0

int

119905119891

0

120593119896(119905119891minus 119904)119860

119896

11198611119906 (119904) 119889119904

+

1198991minus1

sum

119896=0

int

119905119891minus120591

0

120601119896(119905119891minus 120591 minus 119904)119860

119896

11198621119906 (119904) 119889119904

(53)

Therefore 1205961isin ⟨1198601| 1198611 1198621⟩ Moreover we also have

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905119891) + 119873

119896

1198622

119888

119863119896120572

119906 (119905119891minus 120591)]

(54)

which implies that 1205962isin ⟨119873 | 119861

2 1198622⟩ Thus

R (0 0) sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (55)

On the other hand we need to prove

R (0 0) supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (56)

Suppose that

120596 = [1205961

1205962

] isin ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (57)

where 1205961isin ⟨1198601| 1198611 1198621⟩ 1205962isin ⟨119873 | 119861

2 1198622⟩ and 120596

1= 0

1205962

= 0 For the initial conditions 1199090= 0 and 120595(119905) equiv 0 119905 isin

[minus120591 0] let 119906(119904) = 1199061(119904) + 119906

2(119904) then we have

1199091(119905) = int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 1198621[1199061(119904) + 119906

2(119904)] 119889119904

(58)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199061(119905) + 119873

119896

1198622

119888

119863119896120572

1199061(119905 minus 120591)]

minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199062(119905) + 119873

119896

1198622

119888

119863119896120572

1199062(119905 minus 120591)]

(59)

As discussed by Yip and Sincovec [3] we choose 1199061(119904) =

119891(119904)119910(119904) to satisfy

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119891 (119904) 119892 (119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198621119891 (119904 minus 120591) 119892 (119904 minus 120591) 119889119904

= 1205961minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199062(119904) 119889119904

minus int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 11986211199062(119904) 119889119904 equiv

1

119888

119863119896120572

1199061(119905) =

119888

119863119896120572

1199061(119905 minus 120591) = 0 119896 = 0 1 2 120592 minus 1

(60)

Then 1199061(119904) contributes nothing to 119909

2(119905) at 119904 = 119905 and 119904 = 119905 minus 120591

Now we prove that (60) is trueIn fact for

1isin ⟨1198601| 1198611 1198621⟩ fromLemma 2 there exists

119911 isin R119899 such that119882(119891 119905)119911 = 1

Let

119910 (119904) =

119891 (119904) [119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904))

+119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0 le 119904 le 119905 minus 120591

119891 (119904) 119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

119905 minus 120591 le 119904 le 119905 minus120591 le 119904 le 0

(61)

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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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

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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

4 Journal of Applied Mathematics

Applying the inverse Laplace transform we obtain

1199091(119905) = 119864

1205721(1198601119905120572

) 1199091(0)

+ int

119905

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times [1198611119906 (119904) + 119862

1119906 (119904 minus 120591)] 119889119904

= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(23)

On the other hand inserting (16) into both sides of the secondequation in system (11) yields

119873119888

119863120572

1199092(119905) minus 119909

2(119905) minus 119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591)

= minus

]minus1

sum

119896=0

[119873119896+1

1198612

119888

119863(119896+1)120572

119906 (119905) + 119873119896+1

1198622

119888

119863(119896+1)120572

119906 (119905 minus 120591)]

minus 1198612119906 (119905) minus 119862

2119906 (119905 minus 120591)

+

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

= 0

(24)

Then (16) is a solution of the second equation in system (11)Next we will show that any solution of the second

equation in system (11) has the form of (16) Define linearoperator 119879 C1([0 +infin)R1198992) rarr C([0 +infin)R1198992) by1198791199092(119905) =

119888

119863120572

1199092(119905) Then the second equation in system (11)

becomes

(119868 minus 119873119879) 1199092(119905) = minus119861

2119906 (119905) minus 119862

2119906 (119905 minus 120591) (25)

Since 119873 is nilpotent 119873 and 119879 can exchange and (119868 minus

119873119879)(sum]minus1119896=0

119873119896

119879119896

) = 119868 then we have

1199092(119905) = minus(119868 minus 119873119879)

minus1

[1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

+infin

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus(

]minus1

sum

119896=0

119873119896

119879119896

) [1198612119906 (119905) + 119862

2119906 (119905 minus 120591)]

= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)]

(26)

Thus (16) is indeed a solution of the second equation insystem (11)

Therefore the proof of Theorem 1 is completed

For convenience sake some notations are introduced asfollows

Given matrices 119860 isin R119899times119899 and 119861 119862 isin R119899times119898 denote Im(119861)

as the range of 119861 that is

Im (119861) = 119910 | 119910 = 119861119909 forall119909 isin R119898

(27)

Let

⟨119860 | 119861 119862⟩ = Im (119861) + Im (119860119861) + sdot sdot sdot + Im (119860119899minus1

119861)

+ Im (119862) + Im (119860119862) + sdot sdot sdot + Im (119860119899minus1

119862)

(28)

Then the space ⟨1198601| 1198611 1198621⟩ is spanned by the columns of

[1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] (29)

and the space ⟨119873|1198612 1198622⟩ is spanned by the columns of

[1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] (30)

For any nonzero polynomial 119891(119904) we define119882(119891 119905) R1198991 rarr

R1198991 by

119882(119891 119905) 119911

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus119904)

120572minus1

119864120572120572

[119860lowast

1(119905minus 119904)

120572

] 119891 (119904) 119911 119889119904

(31)

where lowast denotes the matrix transposeThe following lemma is required for the main result

which is a natural extension of Lemma 2-11 in [1]

Lemma 2 Given matrices 1198601isin R1198991times1198991 and 119861

1 1198621isin R1198991times1198981

then

Im119882(119891 119905) = ⟨1198601| 1198611 1198621⟩ (32)

Proof The formula (32) holds if and only if

Ker119882(119891 119905) =

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(33)

Journal of Applied Mathematics 5

If 1199110isin Ker119882(119891 119905) and 119911

0= 0 119911lowast0119866(119905)1199110= 0 that is

int

119905

119905minus120591

10038171003817100381710038171003817119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[(119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904

+ int

119905minus120591

0

10038171003817100381710038171003817[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904 = 0

(34)

then we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0= 0 119904 isin [119905 minus 120591 119905]

(35)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119891 (119904) 119911

0= 0 119904 isin [0 119905 minus 120591]

(36)

Since polynomial 119891(119904) only has a finite number of zero on119905 minus 120591 le 119904 le 119905 and 0 le 119904 le 119905 minus 120591 we immediately have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119911

0= 0 119904 isin [119905 minus 120591 119905] (37)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0= 0 119904 isin [0 119905 minus 120591]

(38)

Repeatedly taking the Caputorsquos derivative on both sides of(37) we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] (119860

lowast

1)119894

1199110= 0

119894 = 1 2 1198991minus 1 119904 isin [119905 minus 120591 119905]

(39)

Substituting 119904 = 119905 in (39) yields

119861lowast

(119860lowast

)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (40)

which implies that 1199110isin Ker119861lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

According to Cayley-Hamilton theorem [4] (119905 minus

119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] can be represented as

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] =

+infin

sum

119896=0

(119905 minus 119904)119896120572+120572minus1

Γ (119896120572 + 120572)119860119896

1

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119860

119896

1

(41)

For 119904 isin [0 119905 minus 120591] it follows from (39) and (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(42)

Inserting (42) into (38) yields

119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) 119911

0= 0 (43)

Repeatedly taking the Caputorsquos fractional derivative on bothsides of (43) and letting 119904 = 119905 minus 120591 we have

119862lowast

1119860lowast

1

119896

1199110= 0 119896 = 1 2 119899

1minus 1 (44)

which implies that 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

Therefore we have

Ker119882(119891 119905) sube

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(45)

Conversely if 1199110isin ⋂1198991minus1

119894=0Ker119861lowast1(119860lowast

1)119894

⋂1198991minus1

119894=0Ker119862lowast

1(119860lowast

1)119894

and 1199110

= 0 then 1199110isin Ker119861lowast

1(119860lowast

1)119894 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 =

1 2 1198991minus 1 and

119861lowast

1(119860lowast

1)119894

1199110= 119862lowast

1(119860lowast

1)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (46)

For 119904 isin [119905 minus 120591 119905] it follows from (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(47)

For 119904 isin [0 119905 minus 120591] the same argument yields

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110

+

1198991minus1

sum

119896=0

120582119896(119905 minus 120591 minus 119904) 119862

lowast

1119860lowast

1

119896

1199110= 0

(48)

Hence 1199110isin Ker119882(119891 119905) which implies

Ker119882(119891 119905) supe

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(49)

From (45) and (49) we know that (32) is true The proof istherefore completed

The reachable set for system (1) (or (11)) may be definedas follows

Definition 3 Any vector 120596 isin R119899 in 119899-dimensional vectorspace is said to reachable if there exists an admissibleinitial data (119909

0 120595) isin A admissible control input 119906(119905) isin

Cℎ119901([0 +infin)R119898) and 119905

119891gt 0 such that the solution of system

(1) (or (11)) satisfies 119909(119905119891 1199090 120595) = 120596

Let R(1199090 120595) be the reachable set from any admissible

initial data (1199090 120595) isin A Then

R (1199090 120595)

= 120596 isin R119899

| (1199090 120595) isin A 119906 (119905) isin Cℎ

119901([0 +infin) R

119898

)

119905119891gt 0 119909 (119905

119891 1199090 120595) = 120596

(50)

6 Journal of Applied Mathematics

Considering the reachable set from the initial conditions1199090= 0 and 120595 equiv 0 we have the following theorem

Theorem 4 For system (11) the reachable set R(0 0) can berepresented as

R (0 0) = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (51)

where ldquooplusrdquo is the direct sum in vector space

Proof Let 120596 isin R(0 0) from (15) (16) and (50) then thereexists 119905

119891gt 0 and 119906(119905) isin Cℎ

119901([0 +infin)R119898) such that

1205961= int

119905119891

0

(119905119891minus 119904)120572minus1

119864120572120572

[1198601(119905119891minus 119904)120572

] 1198611119906 (119904) 119889119904

+ int

119905119891minus120591

0

(119905119891minus 120591 minus 119904)

120572minus1

119864120572120572

[1198601(119905119891minus 120591 minus 119904)

120572

]

times 1198621119906 (119904) 119889119904

(52)

Combining (41) and (52) yields

1205961=

1198991minus1

sum

119896=0

int

119905119891

0

120593119896(119905119891minus 119904)119860

119896

11198611119906 (119904) 119889119904

+

1198991minus1

sum

119896=0

int

119905119891minus120591

0

120601119896(119905119891minus 120591 minus 119904)119860

119896

11198621119906 (119904) 119889119904

(53)

Therefore 1205961isin ⟨1198601| 1198611 1198621⟩ Moreover we also have

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905119891) + 119873

119896

1198622

119888

119863119896120572

119906 (119905119891minus 120591)]

(54)

which implies that 1205962isin ⟨119873 | 119861

2 1198622⟩ Thus

R (0 0) sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (55)

On the other hand we need to prove

R (0 0) supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (56)

Suppose that

120596 = [1205961

1205962

] isin ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (57)

where 1205961isin ⟨1198601| 1198611 1198621⟩ 1205962isin ⟨119873 | 119861

2 1198622⟩ and 120596

1= 0

1205962

= 0 For the initial conditions 1199090= 0 and 120595(119905) equiv 0 119905 isin

[minus120591 0] let 119906(119904) = 1199061(119904) + 119906

2(119904) then we have

1199091(119905) = int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 1198621[1199061(119904) + 119906

2(119904)] 119889119904

(58)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199061(119905) + 119873

119896

1198622

119888

119863119896120572

1199061(119905 minus 120591)]

minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199062(119905) + 119873

119896

1198622

119888

119863119896120572

1199062(119905 minus 120591)]

(59)

As discussed by Yip and Sincovec [3] we choose 1199061(119904) =

119891(119904)119910(119904) to satisfy

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119891 (119904) 119892 (119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198621119891 (119904 minus 120591) 119892 (119904 minus 120591) 119889119904

= 1205961minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199062(119904) 119889119904

minus int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 11986211199062(119904) 119889119904 equiv

1

119888

119863119896120572

1199061(119905) =

119888

119863119896120572

1199061(119905 minus 120591) = 0 119896 = 0 1 2 120592 minus 1

(60)

Then 1199061(119904) contributes nothing to 119909

2(119905) at 119904 = 119905 and 119904 = 119905 minus 120591

Now we prove that (60) is trueIn fact for

1isin ⟨1198601| 1198611 1198621⟩ fromLemma 2 there exists

119911 isin R119899 such that119882(119891 119905)119911 = 1

Let

119910 (119904) =

119891 (119904) [119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904))

+119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0 le 119904 le 119905 minus 120591

119891 (119904) 119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

119905 minus 120591 le 119904 le 119905 minus120591 le 119904 le 0

(61)

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

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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

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

Journal of Applied Mathematics 5

If 1199110isin Ker119882(119891 119905) and 119911

0= 0 119911lowast0119866(119905)1199110= 0 that is

int

119905

119905minus120591

10038171003817100381710038171003817119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[(119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904

+ int

119905minus120591

0

10038171003817100381710038171003817[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119891 (119904) 119911

0

10038171003817100381710038171003817

2

119889119904 = 0

(34)

then we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119891 (119904) 119911

0= 0 119904 isin [119905 minus 120591 119905]

(35)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119891 (119904) 119911

0= 0 119904 isin [0 119905 minus 120591]

(36)

Since polynomial 119891(119904) only has a finite number of zero on119905 minus 120591 le 119904 le 119905 and 0 le 119904 le 119905 minus 120591 we immediately have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] 119911

0= 0 119904 isin [119905 minus 120591 119905] (37)

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0= 0 119904 isin [0 119905 minus 120591]

(38)

Repeatedly taking the Caputorsquos derivative on both sides of(37) we have

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)] (119860

lowast

1)119894

1199110= 0

119894 = 1 2 1198991minus 1 119904 isin [119905 minus 120591 119905]

(39)

Substituting 119904 = 119905 in (39) yields

119861lowast

(119860lowast

)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (40)

which implies that 1199110isin Ker119861lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

According to Cayley-Hamilton theorem [4] (119905 minus

119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] can be represented as

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] =

+infin

sum

119896=0

(119905 minus 119904)119896120572+120572minus1

Γ (119896120572 + 120572)119860119896

1

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119860

119896

1

(41)

For 119904 isin [0 119905 minus 120591] it follows from (39) and (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(42)

Inserting (42) into (38) yields

119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) 119911

0= 0 (43)

Repeatedly taking the Caputorsquos fractional derivative on bothsides of (43) and letting 119904 = 119905 minus 120591 we have

119862lowast

1119860lowast

1

119896

1199110= 0 119896 = 1 2 119899

1minus 1 (44)

which implies that 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 = 1 2 119899

1minus 1

Therefore we have

Ker119882(119891 119905) sube

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(45)

Conversely if 1199110isin ⋂1198991minus1

119894=0Ker119861lowast1(119860lowast

1)119894

⋂1198991minus1

119894=0Ker119862lowast

1(119860lowast

1)119894

and 1199110

= 0 then 1199110isin Ker119861lowast

1(119860lowast

1)119894 1199110isin Ker119862lowast

1(119860lowast

1)119894 119894 =

1 2 1198991minus 1 and

119861lowast

1(119860lowast

1)119894

1199110= 119862lowast

1(119860lowast

1)119894

1199110= 0 119894 = 1 2 119899

1minus 1 (46)

For 119904 isin [119905 minus 120591 119905] it follows from (41) that

119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110= 0

(47)

For 119904 isin [0 119905 minus 120591] the same argument yields

[119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) + 119862

lowast

1(119905 minus 120591 minus 119904)

120572minus1

times119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904)) ] 119911

0

=

1198991minus1

sum

119896=0

120574119896(119905 minus 119904) 119861

lowast

1119860lowast

1

119896

1199110

+

1198991minus1

sum

119896=0

120582119896(119905 minus 120591 minus 119904) 119862

lowast

1119860lowast

1

119896

1199110= 0

(48)

Hence 1199110isin Ker119882(119891 119905) which implies

Ker119882(119891 119905) supe

1198991minus1

⋂

119894=0

Ker119861lowast1(119860lowast

1)119894

1198991minus1

⋂

119894=0

Ker119862lowast1(119860lowast

1)119894

(49)

From (45) and (49) we know that (32) is true The proof istherefore completed

The reachable set for system (1) (or (11)) may be definedas follows

Definition 3 Any vector 120596 isin R119899 in 119899-dimensional vectorspace is said to reachable if there exists an admissibleinitial data (119909

0 120595) isin A admissible control input 119906(119905) isin

Cℎ119901([0 +infin)R119898) and 119905

119891gt 0 such that the solution of system

(1) (or (11)) satisfies 119909(119905119891 1199090 120595) = 120596

Let R(1199090 120595) be the reachable set from any admissible

initial data (1199090 120595) isin A Then

R (1199090 120595)

= 120596 isin R119899

| (1199090 120595) isin A 119906 (119905) isin Cℎ

119901([0 +infin) R

119898

)

119905119891gt 0 119909 (119905

119891 1199090 120595) = 120596

(50)

6 Journal of Applied Mathematics

Considering the reachable set from the initial conditions1199090= 0 and 120595 equiv 0 we have the following theorem

Theorem 4 For system (11) the reachable set R(0 0) can berepresented as

R (0 0) = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (51)

where ldquooplusrdquo is the direct sum in vector space

Proof Let 120596 isin R(0 0) from (15) (16) and (50) then thereexists 119905

119891gt 0 and 119906(119905) isin Cℎ

119901([0 +infin)R119898) such that

1205961= int

119905119891

0

(119905119891minus 119904)120572minus1

119864120572120572

[1198601(119905119891minus 119904)120572

] 1198611119906 (119904) 119889119904

+ int

119905119891minus120591

0

(119905119891minus 120591 minus 119904)

120572minus1

119864120572120572

[1198601(119905119891minus 120591 minus 119904)

120572

]

times 1198621119906 (119904) 119889119904

(52)

Combining (41) and (52) yields

1205961=

1198991minus1

sum

119896=0

int

119905119891

0

120593119896(119905119891minus 119904)119860

119896

11198611119906 (119904) 119889119904

+

1198991minus1

sum

119896=0

int

119905119891minus120591

0

120601119896(119905119891minus 120591 minus 119904)119860

119896

11198621119906 (119904) 119889119904

(53)

Therefore 1205961isin ⟨1198601| 1198611 1198621⟩ Moreover we also have

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905119891) + 119873

119896

1198622

119888

119863119896120572

119906 (119905119891minus 120591)]

(54)

which implies that 1205962isin ⟨119873 | 119861

2 1198622⟩ Thus

R (0 0) sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (55)

On the other hand we need to prove

R (0 0) supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (56)

Suppose that

120596 = [1205961

1205962

] isin ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (57)

where 1205961isin ⟨1198601| 1198611 1198621⟩ 1205962isin ⟨119873 | 119861

2 1198622⟩ and 120596

1= 0

1205962

= 0 For the initial conditions 1199090= 0 and 120595(119905) equiv 0 119905 isin

[minus120591 0] let 119906(119904) = 1199061(119904) + 119906

2(119904) then we have

1199091(119905) = int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 1198621[1199061(119904) + 119906

2(119904)] 119889119904

(58)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199061(119905) + 119873

119896

1198622

119888

119863119896120572

1199061(119905 minus 120591)]

minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199062(119905) + 119873

119896

1198622

119888

119863119896120572

1199062(119905 minus 120591)]

(59)

As discussed by Yip and Sincovec [3] we choose 1199061(119904) =

119891(119904)119910(119904) to satisfy

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119891 (119904) 119892 (119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198621119891 (119904 minus 120591) 119892 (119904 minus 120591) 119889119904

= 1205961minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199062(119904) 119889119904

minus int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 11986211199062(119904) 119889119904 equiv

1

119888

119863119896120572

1199061(119905) =

119888

119863119896120572

1199061(119905 minus 120591) = 0 119896 = 0 1 2 120592 minus 1

(60)

Then 1199061(119904) contributes nothing to 119909

2(119905) at 119904 = 119905 and 119904 = 119905 minus 120591

Now we prove that (60) is trueIn fact for

1isin ⟨1198601| 1198611 1198621⟩ fromLemma 2 there exists

119911 isin R119899 such that119882(119891 119905)119911 = 1

Let

119910 (119904) =

119891 (119904) [119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904))

+119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0 le 119904 le 119905 minus 120591

119891 (119904) 119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

119905 minus 120591 le 119904 le 119905 minus120591 le 119904 le 0

(61)

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

6 Journal of Applied Mathematics

Considering the reachable set from the initial conditions1199090= 0 and 120595 equiv 0 we have the following theorem

Theorem 4 For system (11) the reachable set R(0 0) can berepresented as

R (0 0) = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (51)

where ldquooplusrdquo is the direct sum in vector space

Proof Let 120596 isin R(0 0) from (15) (16) and (50) then thereexists 119905

119891gt 0 and 119906(119905) isin Cℎ

119901([0 +infin)R119898) such that

1205961= int

119905119891

0

(119905119891minus 119904)120572minus1

119864120572120572

[1198601(119905119891minus 119904)120572

] 1198611119906 (119904) 119889119904

+ int

119905119891minus120591

0

(119905119891minus 120591 minus 119904)

120572minus1

119864120572120572

[1198601(119905119891minus 120591 minus 119904)

120572

]

times 1198621119906 (119904) 119889119904

(52)

Combining (41) and (52) yields

1205961=

1198991minus1

sum

119896=0

int

119905119891

0

120593119896(119905119891minus 119904)119860

119896

11198611119906 (119904) 119889119904

+

1198991minus1

sum

119896=0

int

119905119891minus120591

0

120601119896(119905119891minus 120591 minus 119904)119860

119896

11198621119906 (119904) 119889119904

(53)

Therefore 1205961isin ⟨1198601| 1198611 1198621⟩ Moreover we also have

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905119891) + 119873

119896

1198622

119888

119863119896120572

119906 (119905119891minus 120591)]

(54)

which implies that 1205962isin ⟨119873 | 119861

2 1198622⟩ Thus

R (0 0) sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (55)

On the other hand we need to prove

R (0 0) supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (56)

Suppose that

120596 = [1205961

1205962

] isin ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (57)

where 1205961isin ⟨1198601| 1198611 1198621⟩ 1205962isin ⟨119873 | 119861

2 1198622⟩ and 120596

1= 0

1205962

= 0 For the initial conditions 1199090= 0 and 120595(119905) equiv 0 119905 isin

[minus120591 0] let 119906(119904) = 1199061(119904) + 119906

2(119904) then we have

1199091(119905) = int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611[1199061(119904) + 119906

2(119904)] 119889119904

+ int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 1198621[1199061(119904) + 119906

2(119904)] 119889119904

(58)

1199092(119905) = minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199061(119905) + 119873

119896

1198622

119888

119863119896120572

1199061(119905 minus 120591)]

minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

1199062(119905) + 119873

119896

1198622

119888

119863119896120572

1199062(119905 minus 120591)]

(59)

As discussed by Yip and Sincovec [3] we choose 1199061(119904) =

119891(119904)119910(119904) to satisfy

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119891 (119904) 119892 (119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198621119891 (119904 minus 120591) 119892 (119904 minus 120591) 119889119904

= 1205961minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199062(119904) 119889119904

minus int

119905minus120591

0

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

]

times 11986211199062(119904) 119889119904 equiv

1

119888

119863119896120572

1199061(119905) =

119888

119863119896120572

1199061(119905 minus 120591) = 0 119896 = 0 1 2 120592 minus 1

(60)

Then 1199061(119904) contributes nothing to 119909

2(119905) at 119904 = 119905 and 119904 = 119905 minus 120591

Now we prove that (60) is trueIn fact for

1isin ⟨1198601| 1198611 1198621⟩ fromLemma 2 there exists

119911 isin R119899 such that119882(119891 119905)119911 = 1

Let

119910 (119904) =

119891 (119904) [119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904))

+119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 120591 minus 119904))] 119911

0 le 119904 le 119905 minus 120591

119891 (119904) 119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

(119860lowast

1(119905 minus 119904)) 119911

119905 minus 120591 le 119904 le 119905 minus120591 le 119904 le 0

(61)

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

Journal of Applied Mathematics 7

Then

int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986111199061(119904) 119889119904

+ int

119905

120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 11986211199061(119904 minus 120591) 119889119904

= int

119905

119905minus120591

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

]

times 1198611119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

+ int

119905minus120591

0

119891 (119904) (119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621

times 119862lowast

1(119905 minus 120591 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 120591 minus 119904)

120572

]

+119861lowast

1(119905 minus 119904)

120572minus1

119864120572120572

[119860lowast

1(119905 minus 119904)

120572

] 119891 (119904) 119911 119889119904

= 119882(119891 119905) 119911 = 1

(62)

Therefore (60) is trueFor 1205962isin ⟨119873 | 119861

2 1198622⟩ there exist 119910

119896 119911119896such that

1205962= minus

]minus1

sum

119896=0

119873119896

1198612119910119896minus

]minus1

sum

119896=0

119873119896

1198622119911119896 (63)

There exists a function 119901(119904) such that 119888119863119896120572119901(0) = 0119888

119863119896120572

119901(119905) = 119910119896 and 119888119863119896120572119901(119905 minus 120591) = 119911

119896 Let

1199062(119904) =

119901 (119904) 0 le 119904 le 119905

0 minus120591 le 119904 le 0(64)

then we have

1199092(119905) = minus

]minus1

sum

119896=0

119873119896

1198612

119888

119863119896120572

1199062(119905)

minus

]minus1

sum

119896=0

119873119896

1198622

119888

119863119896120572

1199062(119905 minus 120591) = 120596

2

(65)

and 1199092(0) = 0 It follows from (60) and (65) that 120596 isin R(0 0)

Thus (56) holds Combining (55) and (56) we know that (51)is true Then the proof of Theorem 4 is completed

4 Controllability Criteria

In this section we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach

Definition 5 System (1) (or (11)) is said to be controllable at119905119891gt 0 if one can reach any state at 119905

119891from any admissible

initial data (1199090 120595) isin A

Theorem 6 Canonical system (11) is controllable if and only if

rank [1198611 11986011198611 119860

1198991minus1

11198611 1198621 11986011198621 119860

1198991minus1

11198621] = 1198991

(66)

rank [1198612 1198731198612 119873

1198992minus1

1198612 1198622 1198731198622 119873

1198992minus1

1198622] = 1198992

(67)

Proof We firstly prove the necessity of Theorem 6 If system(11) is controllable for any 120596 = [

1205961

1205962] isin R119899 to the initial state

1199090= 0 and initial control function 120595 equiv 0 there exists 119905

119891gt 0

and a control function 119906(sdot) isin Cℎ119901([0 +infin) such that 120596

1 1205962

could be written in the form of (58) and (59) That is120596 isin ⟨119860

1| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (68)

which implies thatR119899

sube ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (69)

ObviouslyR119899

supe ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ (70)

Consequently we have

R119899

= ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩

⟨1198601| 1198611 1198621⟩ = R

1198991 ⟨119873 | 119861

2 1198622⟩ = R

1198992

(71)

Thus (66) and (67) are trueNext we prove the sufficiency of Theorem 6 If (66) and

(67) are true then we know that (71) holds For any 120596 isin R119899

and any initial state 1199090and initial control function 120595(sdot) let

1198961= 1205961minus 1198641205721

(1198601119905120572

) 1199091(0)

minus int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611120595 (0) 119889119904

minus int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621 120595 (0) 119889119904

minus int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

1198962= 1205962+ 1198612120595 (0) + 119862

2120595 (0)

(72)

For 119896 = [1198961

1198962

] isin R119899 = ⟨1198601| 1198611 1198621⟩ oplus ⟨119873 | 119861

2 1198622⟩ we have

119896 isin R(0 0) Then there exists a control (119904) such that

1198961= int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611 (119904) 119889119904

+ int

119905minus120591

0

[(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611+ (119905 minus 120591 minus 119904)

120572minus1

times 119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621] (119904) 119889119904

1198962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

(119905) + 119873119896

1198622

119888

119863119896120572

(119905 minus 120591)]

(73)

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

8 Journal of Applied Mathematics

Let 119906(119904) = (119904) + 120595(0) then we have

1205961= 1198641205721

(1198601119905120572

) 1199091(0)

+ int

119905

119905minus120591

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611119906 (119904) 119889119904

+ int

119905minus120591

0

(119905 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 119904)

120572

] 1198611

+(119905minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905minus 120591 minus119904)

120572

] 1198621 119906 (119904) 119889119904

+ int

0

minus120591

(119905 minus 120591 minus 119904)120572minus1

119864120572120572

[1198601(119905 minus 120591 minus 119904)

120572

] 1198621120595 (119904) 119889119904

(74)

1205962= minus

]minus1

sum

119896=0

[119873119896

1198612

119888

119863119896120572

119906 (119905) + 119873119896

1198622

119888

119863119896120572

119906 (119905 minus 120591)] (75)

Thus by Definition 5 system (11) is controllable Thereforethe proof is completed

Applying the results of Yip and Sincovec [3] Tang and Li[4] andTheorem 6 to the pair of matrices we can obtain thefollowing results

Theorem 7 Canonical system (11) is controllable if and only if

rank [119904119868 minus 1198601 1198611 1198621] = 1198991 forall119904 isin C 119904 finite

rank [119873 1198612 1198622] = 1198992

(76)

Theorem 8 System (1) is controllable if and only if

rank [119904119864 minus 119860 119861 119862] = 119899 forall119904 isin C 119904 finite

rank [119864 119861 119862] = 119899

(77)

Remark 9 If we choose 120572 = 1 and 119862 = 0 then system (1)reduces to an integer-order singular system without controldelay FromTheorem 8 system (1) is controllable if and onlyif

rank [119904119864 minus 119860 119861] = 119899 forall119904 isin C 119904 finite

rank [119864 119861] = 119899

(78)

which is just the result in [1]

If we choose 119864 = 119868 119862 = 0 then system (1) reducesto a fractional normal system without control delay FromTheorem 8 system (1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861] = 119899 (79)

which is just Corollary 34 in [23]If we choose 119864 = 119868 then system (1) reduces to a fractional

normal system with control delay From Theorem 8 system(1) is controllable if and only if

rank [119861 119860119861 119860119899minus1119861 119862 119860119862 119860119899minus1119862] = 119899 (80)

which is justTheorem 3 in [25]Therefore our results extendthe existing results [1 23 25]

Remark 10 Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems which is concise and convenient to check thecontrollability of such systems Motivated and inspired bythe works of Sakthivel et al [32ndash35] the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work

5 An Illustrative Example

In this section we give an example to illustrate the applica-tions of Theorem 8

Example 11 Consider the controllability of fractional singu-lar dynamical systems as follows

[

[

1 0 0

0 0 1

0 0 0

]

]

11986312

119909 (119905) = [

[

1 0 0

0 1 1

1 0 1

]

]

119909 (119905) + [

[

1 0

1 1

0 1

]

]

119906 (119905)

+ [

[

1 0

0 1

1 1

]

]

119906(119905 minus120587

4) 119905 isin [0 +infin)

119906 (119905) = 120595 (119905) 119905 isin [minus120587

4 0]

119909 (0) = 1199090

(81)

where the admissible initial data (1199090 120595) isin A Now we apply

Theorem 8 to prove that system (81) is controllable Let ustake

120572 =1

2 119864 = [

[

1 0 0

0 0 1

0 0 0

]

]

119860 = [

[

1 0 0

0 1 1

1 0 1

]

]

119861 = [

[

1 0

1 1

0 1

]

]

119862 = [

[

1 0

0 1

1 1

]

]

(82)

If we choose 120582 = 1 such that det(120582119864 + 119860) = 2 = 0 then (119864 119860)

is regular By using the elementary transformation of matrixone can obtain

rank [119904119864 minus 119860 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 minus1 ⋆ ⋆ ⋆ ⋆

0 0 2 ⋆ ⋆ ⋆ ⋆

]

]

= 3

rank [119864 119861 119862] = rank[

[

1 0 1 ⋆ ⋆ ⋆ ⋆

0 1 0 ⋆ ⋆ ⋆ ⋆

0 0 1 ⋆ ⋆ ⋆ ⋆

]

]

= 3

(83)

Thus byTheorem 8 system (81) is controllable

6 Conclusions

In this paper the reachability and controllability of frac-tional singular dynamical systems with control delay have

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

Journal of Applied Mathematics 9

been investigated The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach An example is also presented to illustrate the effec-tiveness and applicability of the results obtained Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay Comparing with some existing results [1 2325] we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems Motivated and inspired by the worksof Sakthivel et al [32ndash35] the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work

Conflicts of Interests

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

Acknowledgments

The authors are very grateful to the Associate Editor Profes-sorHanHChoi and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper This work is jointly supported by theNational Natural Science Foundation of China under Grantnos 11071001 11072059 and 61272530 the Natural ScienceFoundation of Jiangsu Province of China under Grant noBK2012741 and the Programs of Educational Commissionof Anhui Province of China under Grant nos KJ2011A197and KJ2013Z186 This work is also funded by the Deanshipof Scientific Research King Abdulaziz University (KAU)under Grant no 3-1301434HiCi Therefore the authorsacknowledge technical and financial support of KAU

References

[1] L Dai Singular Control Systems vol 118 of Lecture Notes in Con-trol and Information Sciences Springer Berlin Germany 1989

[2] P Kunkel and V Mehrmann Differential-Algebraic EquationsAnalysis and Numerical Solution EuropeanMathematical Soci-ety Zurich Switzerland 2006

[3] E L Yip and R F Sincovec ldquoSolvability controllability andobservability of continuous descriptor systemsrdquo IEEE Transac-tions on Automatic Control vol 26 no 3 pp 702ndash707 1981

[4] W S Tang and G Q Li ldquoThe criterion for controllability andobservability of singular systemsrdquo Acta Automatica Sinica vol21 no 1 pp 63ndash66 1995

[5] W Jiang and W Z Song ldquoControllability of singular systemswith control delayrdquo Automatica vol 37 no 11 pp 1873ndash18772001

[6] S L Campbell and V H Linh ldquoStability criteria for differential-algebraic equations with multiple delays and their numericalsolutionsrdquo Applied Mathematics and Computation vol 208 no2 pp 397ndash415 2009

[7] H Wang A Xue and R Lu ldquoAbsolute stability criteria for aclass of nonlinear singular systems with time delayrdquo NonlinearAnalysisTheory Methods amp Applications vol 70 no 2 pp 621ndash630 2009

[8] J Xi F Meng Z Shi and Y Zhong ldquoDelay-dependent admis-sible consensualization for singular time-delayed swarm sys-temsrdquo Systems amp Control Letters vol 61 no 11 pp 1089ndash10962012

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

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

[11] K Diethelm The Analysis of Fractional Differential EquationsAn Application-Oriented Exposition Using Differential Operatorsof Caputo Type vol 2004 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[12] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-HollandMathematics Studies Elsevier Science BV Ams-terdam The Netherlands 2006

[13] M Caputo ldquoLinear model of dissipation whose Q is almostfrequency independent IIrdquo Geophysical Journal of the RoyalAstronomical Society vol 13 no 5 pp 529ndash539 1967

[14] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications vol 69 no 8 pp 2677ndash2682 2008

[15] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[16] Z X Zhang andW Jiang ldquoSome results of the degenerate frac-tional differential systemwith delayrdquoComputers ampMathematicswith Applications vol 62 no 3 pp 1284ndash1291 2011

[17] K Sayevand A Golbabai and A Yildirim ldquoAnalysis of dif-ferential equations of fractional orderrdquo Applied MathematicalModelling vol 36 no 9 pp 4356ndash4364 2012

[18] H Zhang J Cao and W Jiang ldquoGeneral solution of linearfractional neutral differential difference equationsrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 4895217 pages 2013

[19] M I Krastanov ldquoOn the constrained small-time controllabilityof linear systemsrdquo Automatica vol 44 no 9 pp 2370ndash23742008

[20] S Zhao and J Sun ldquoA geometric approach for reachability andobservability of linear switched impulsive systemsrdquo NonlinearAnalysis Theory Methods amp Applications vol 72 no 11 pp4221ndash4229 2010

[21] R Manzanilla L G Marmol and C J Vanegas ldquoOn thecontrollability of a differential equation with delayed andadvanced argumentsrdquo Abstract and Applied Analysis vol 2010Article ID 307409 16 pages 2010

[22] R M Du and C P Wang ldquoNull controllability of a class ofsystems governed by coupled degenerate equationsrdquo AppliedMathematics Letters vol 26 no 1 pp 113ndash119 2013

[23] T L Guo ldquoControllability and observability of impulsive frac-tional linear time-invariant systemrdquo Computers amp Mathematicswith Applications vol 64 no 10 pp 3171ndash3182 2012

[24] H Zhang J Cao and W Jiang ldquoControllability criteria for lin-ear fractional differential systemswith state delay and impulsesrdquoJournal of Applied Mathematics vol 2013 Article ID 146010 9pages 2013

10 Journal of Applied Mathematics

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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

[25] W Jiang ldquoThe controllability of fractional control systems withcontrol delayrdquoComputers ampMathematics with Applications vol64 no 10 pp 3153ndash3159 2012

[26] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications vol 74 no 17 pp 5929ndash5942 2011

[27] K Balachandran J Kokila and J J Trujillo ldquoRelative control-lability of fractional dynamical systems with multiple delays incontrolrdquo Computers amp Mathematics with Applications vol 64no 10 pp 3037ndash3045 2012

[28] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 75 no 4 pp 1919ndash19262012

[29] K Balachandran Y Zhou and J Kokila ldquoRelative controllabil-ity of fractional dynamical systemswith delays in controlrdquoCom-munications inNonlinear Science andNumerical Simulation vol17 no 9 pp 3508ndash3520 2012

[30] 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

[31] R Ganesh R Sakthivel N I Mahmudov and S M AnthonildquoApproximate controllability of fractional integrodifferentialevolution equationsrdquo Journal of Applied Mathematics vol 2013Article ID 291816 7 pages 2013

[32] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[33] 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

[34] R Sakthivel and Y Ren ldquoApproximate controllability of frac-tional differential equations with state-dependent delayrdquoResultsin Mathematics vol 63 no 3 pp 949ndash963 2013

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