Research Article Finite-Time Stability of Neutral Fractional Time …downloads.hindawi.com/journals/aaa/2014/610547.pdf · Finite-Time Stability of Neutral Fractional Time-Delay Systems

Research ArticleFinite-Time Stability of Neutral Fractional Time-Delay Systemsvia Generalized Gronwalls Inequality

Pang Denghao and Jiang Wei

School of Mathematical Sciences Anhui University Hefei 230039 China

Correspondence should be addressed to Jiang Wei jiangwei89018126com

Received 22 October 2013 Accepted 2 January 2014 Published 23 February 2014

Academic Editor Irena Rachunkova

Copyright copy 2014 P Denghao and J Wei This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper studies the finite-time stability of neutral fractional time-delay systems With the generalized Gronwall inequalitysufficient conditions of the finite-time stability are obtained for the particular class of neutral fractional time-delay systems

1 Introduction

In this paper we consider a neutral fractional time-delaysystem119888

D120572

119909 (119905) = 119860119909 (119905) + 119861119909 (119905 minus 120591) + 119862119888

D120572

119909 (119905 minus 120591)

119905 isin [0 119879]

119909 (119905) = 120593 (119905) 119905 isin [minus120591 0]

(1)

where 119888D120572 denotes the Caputo fractional derivative of order0 lt 120572 le 1 the vector function 119909(119905) isin 119877

119899 119860 119861 119862are constant system matrices of appropriate dimensions theconstant parameter 120591 gt 0 represents the delay argument and120593(119905) is a given continuously differentiable function on [minus120591 0]

The neutral time-delay systems have received increasingattention (see [1ndash5]) due to their successful applications inpopulation ecology distributed networks containing loss-less transmission lines heat exchangers robots in contactwith rigid environments partial element equivalent circuit(PEEC) the control of constrained manipulators with time-delaymeasurements the systemswhich need the informationof the past state variables and so on

Recently with the development of theories of fractionaldifferential equations (see [6ndash9]) there has been a surge in thestudy of neutral fractional time-delay systems (see [10ndash12])In particular the problem of stability analysis of such systemshas been one of the most interesting topics in control theory

because stability analysis is one of the most important issuesfor control systems (see [13ndash16]) But stability of these systemsproves to be amore complex issue because the systems involvethe derivative of the time-delayed state and the existence oftime-delay is frequently the source of instability althoughthis problem has been investigated for time-delay systemsover many years In the previous literatures many scholarshave utilized the Lyapunov technique characteristic equationmethod state solution approach or Gronwallrsquos approach toderive sufficient conditions for stability of the systems In thispapermotivated by the papers [17 18] we discuss the stabilityof the neutral fractional system with delay via generalizedGronwallrsquos approach

The organization of this paper is as follows In Section 2we summarize some notations and give preliminary resultswhich will be used in this paper In Section 3 we present ourmain results

2 Preliminaries and Lemmas

Let us start with some definitions and lemmas which are usedthroughout this paper

Definition 1 (see [7]) Eulerrsquos gamma function is defined as

Γ (119911) = int

infin

0

119890minus119905

119905119911minus1

119889119905 119911 isin C (2)

where C denotes the complex plane

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 610547 4 pageshttpdxdoiorg1011552014610547

2 Abstract and Applied Analysis

Remark 2 (see [7]) (i) Γ(119911 + 1) = 119911Γ(119911) 119911 isin C and for 119899 isin

Z+ Γ(119899 + 1) = 119899(119899 minus 1) = 119899(ii) Γ(119911)Γ(119911 + (12)) = radic1205872

1minus2119911

Γ(2119911) 2119911 = 0 minus1 minus2 (ii) Γ(119899 + (12)) = radic120587Γ(2119899 + 1)2

2119899

Γ(119899 + 1) = radic120587(2119899)

22119899

119899

Definition 3 (see [7]) The fractional integral of order 120572 withthe lower limit zero for any function 119891(119905) isin 119862([0 +infin) 119877)119905 ge 0 is defined as

119868120572

119891 (119905) = limℎrarr0

119899ℎ=119905

ℎ120572

119899

sum

119903=0

[

120572

119903]119891 (119905 minus 119903ℎ)

=

1

Γ (120572)

int

119905

0

(119905 minus 120579)120572minus1

119891 (120579) 119889120579 120572 gt 0

(3)

where [ 120572119903] = 120572(120572 + 1) sdot sdot sdot (120572 + 119903 minus 1)119903 and Γ(sdot) is the gamma

function

Definition 4 (see [7]) The Riemann-Liouville derivative oforder 120572 with the lower limit zero for any function 119891(119905) isin

119862([0 +infin) 119877) 119905 ge 0 is defined as

119897

D120572

119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

(119905 minus 120579)119899minus120572minus1

119891 (120579) 119889120579

119899 minus 1 lt 120572 lt 119899

(4)

Definition 5 (see [7]) The Caputo derivative of order 120572 forany function 119891(119905) isin 119862

119899

([0 +infin) 119877) 119905 ge 0 is defined as

119888D120572119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0


119891(119899)

(120579) 119889120579

= 119868119899minus120572

119891 (119905) 119899 minus 1 lt 120572 lt 119899

(5)

Remark 6 (see [7]) (i) If a function 119891(119905) isin 119862119899

([0 +infin) 119877)119905 ge 0 then 119888D120572119891(119905) =

119897D120572119891(119905) minus sum119899minus1

119896=0(119891(119896)

(0)Γ(119896 minus 120572 +

1))119905119896minus120572(ii) 119888D120572119891(119905) = 0 119891(119905) = 119862 and 119862 is a constant

Definition 7 (see [9]) The Mittag-Leffler function in twoparameters is defined as

119864120572120573

(119911) =

infin

sum

119896=0

(119911)119896

Γ (120572119896 + 120573)

119911 isin C (6)

where 120572 gt 0 120573 gt 0 and 119911 isin C

Remark 8 (see [9]) (i) For 120573 = 1 1198641205721(120582119911120572

) = 119864120572(120582119911120572

) =

suminfin

119896=0(120582119896

(119911120572

)119896

Γ(120572119896 + 1)) and 119864(11)

(119911) = 119890119911 119911 isin C

(ii) For120573 = 1 thematrix extension of the aforementionedMittag-Leffler function has the following representation119864120572(119860119905120572

) = suminfin

119896=0((119860119896

(119905120572

)119896

Γ(120572119896 + 1)) 119911 isin C and119888D120572119864

120572(119860119905120572

) = 119860119864120572(119860119905120572

)

Lemma 9 (see [19] generalized Gronwallrsquos inequality) Sup-pose 119909(119905) 119886(119905) are nonnegative and local integrable on 0 le 119905 lt

119879 some 119879 le infin and 119892(119905) is a nonnegative nondecreasing

continuous function defined on 0 le 119905 lt 119879 119892(119905) le 119872 =

constant 120572 gt 0 with

119909 (119905) le 119886 (119905) + 119892 (119905) int

119905

0

(119905 minus 119904)120572minus1

119909 (119904) 119889119904 (7)

on this interval Then

119909 (119905) le 119886 (119905) + 119892 (119905) int

119905

0

[

infin

sum

119899=1

(119892 (119905) Γ (120572))119899

Γ (119899120572)

(119905 minus 119904)119899120572minus1

119886 (119904)] 119889119904

0 le 119905 lt 119879

(8)

Lemma 10 (see [19]) Under the hypothesis of Theorem 13 let119886(119905) be a nondecreasing function on [0 119879) Then

119909 (119905) le 119886 (119905) 119864120572(119892 (119905) Γ (120572) 119905

120572

) (9)

where 119864120572is the Mittag-Leffler function

3 Main Results

In this section we discuss some problems of the neutralfractional time-delay system (1)

Let us denote by 119862([119886 119887]) the space of all continuous realfunctions defined on [119886 119887] and by 119862([119886 119887] 119877

119899

) the Banachspace of continuous functions mapping the interval [119886 119887]into 119877

119899 with the topology of uniform convergence Let 119862 =

119862([minus120591 0] 119877119899

) [119886 119887] = [minus120591 0] and designate the norm of anelement 120593 in 119862 by

10038171003817100381710038171205931003817100381710038171003817= supminus120591le119905le0

1003817100381710038171003817120593 (119905)

1003817100381710038171003817 (10)

Let 119883 = 119862([minus120591 119879] 119877119899

) and 119909(119905) = 120593(119905) 119905 isin [minus120591 0] beequipped with the norm

119909 (119905) = sup0le119905le119879

119909 (119905)

1003817100381710038171003817119909119905

1003817100381710038171003817= 119909 (119905 + 120579) = sup

minus120591le120579le0

119909 (119905 + 120579)

forall119909 isin 119883

(11)

where obviously 119909(119905) le 119909119905

Let ]max(sdot) be the largest singular value of matrix (sdot)namely

]119860119861max = ]max (119860) + ]max (119861) (12)

For convenience we denote ]max(119860) by ]0 ]max(119861) by ]

1

]max(119862) by ]2 ]max(119863) by ]

3 and ]max(119860119861) by ]

01 respec-

tively

Definition 11 (see [18]) The system given by (1) and satisfyinginitial condition 119909(119905) = 120593(119905) for 119905 isin [minus120591 0] is finite stablewith respect to 119905

0 120575 120598 119869 120575 lt 120598 119869 = [119905

0 1199050+ 119879] if and only if

10038171003817100381710038171205931003817100381710038171003817lt 120575 (13)

implies

119909 (119905) lt 120598 forall119905 isin 119869 (14)

Abstract and Applied Analysis 3

Theorem 12 If 119909(119905) = 119909(119905 120593) is a solution of the systems (1)then there exists a positive constant ] such that

119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) forall119905 isin 119869 = [0 119879]

(15)

Proof According to the properties of the fractional order 0 lt

120572 lt 1 one can obtain a solution in the form of the equivalentVolterra integral equation [12]

119909 (119905) = 119909 (0) + 119862 (119909 (119905 minus 120591) minus 119909 (minus120591))

+

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

(119860119909 (119904) + 119861119909 (119904 minus 120591)) 119889119904

= 120593 (0) + 119862120593 (minus120591) + 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

(119860119909 (119904) + 119861119909 (119904 minus 120591)) 119889119904 119905 ge 0

(16)

Using appropriate property of the norm (sdot) in (16) andapplying (10) it follows that

119909 (119905) le1003817100381710038171003817120593 (0)

1003817100381710038171003817+1003817100381710038171003817119862120593 (minus120591)

1003817100381710038171003817+ 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860119909 (119904) + 119861119909 (119904 minus 120591) 119889119904

le (1 + 119862)10038171003817100381710038171205931003817100381710038171003817+ 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860119909 (119904) + 119861119909 (119904 minus 120591) 119889119904

le (1 + 119862)10038171003817100381710038171205931003817100381710038171003817+ 119862 119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860 119909 (119904) 119889119904

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119861 119909 (119904 minus 120591) 119889119904

le (1 + ]2)10038171003817100381710038171205931003817100381710038171003817+ ]2119909 (119905 minus 120591)

+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904) 119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904 minus 120591) 119889119904 119905 ge 0

(17)

For 0 le 119905 le 120591 119909(119905 minus 120591) le 120593 (17) can be rewritten as

119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904) 119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904 minus 120591) 119889119904 0 le 119905 le 120591

(18)

From Definition 3 we can see 119868120572

119891(119905) is an increasingfunction of 119905 if 119891(119905) gt 0 So (]

0Γ(120572)) int

119905

0

|119905minus 119904|120572minus1

119909(119904)119889119904 and

(]1Γ(120572)) int

119905

0

|119905minus119904|120572minus1

119909(119904minus120591)119889119904 are both increasing functionswith regard to 119905 Taking into account (18) and (11) it yields that

1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817+

]01

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

0 le 119905 le 120591

(19)

Let us introduce a function 119886(119905) such as

119886 (119905) = (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817 (20)

where the function 119886(119905) is nondecreasing apparentlyNow with the corollary of the generalized Gronwall

inequality (9) we can obtain1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]01119905120572

) 0 le 119905 le 120591 (21)

Similarly the same argument implies the following esti-mate

1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)

100381710038171003817100381710038171199091205910

10038171003817100381710038171003817119864120572(]01(119905 minus 1205910)120572

)

1205910le 119905 le 120591

0+ 120591 120591

0ge 0

(22)

From Definition 7 we know that the Mittag-Lefflerfunction 119864

120572(119905) is an increasing function with regard to 119905

Therefore there exists ] gt ]01such that 119864

120572(]120591120572) gt 119864

120572(]01120591120572

)

and 119864120572(](119905 minus 120591)

120572

)119864120572(]01120591120572

)119864120572(]119905120572) lt 1(1 + 2]

2)

Relationships (21) and (22) suggest the following generalexpression

1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) 0 le 119905 le 119899120591 le 119879

(23)

To prove formula (23) by induction we have to show thatit holds for 119899 = 1 because of formula (21) and if it holds for 119899 =

119896 then it holds also for 119899 = 119896+ 1 Indeed for 119905 isin [120591 (119896 + 1)120591]119905 minus 120591 isin [0 119896120591] on the one hand using formula (22)we have

1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)1003817100381710038171003817119909119905minus120591

1003817100381710038171003817119864120572(]01120591120572

) (24)

On the other hand using formula (23) we obtain1003817100381710038171003817119909119905minus120591

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(](119905 minus 120591)

120572

) (25)

Taking into account (24) and (25) we conclude that1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2) [(1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(](119905 minus 120591)

120572

)] 119864120572(]01120591120572

)

= (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572)

times

(1 + 2]2) 119864120572(](119905 minus 120591)

120572

) 119864120572(]01120591120572

)

119864120572(]119905120572)

le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572)

(26)


That is

119909 (119905) le1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (27)

The proof is completed

Theorem 13 The neutral fractional time-delay systems givenby (1) are finite-time stable with respect to 0 120575 120598 119869 120575 lt 120598 ifthe following condition is satisfied

(1 + 2]2) 119864120572(]119905120572) le

120598

120575

forall119905 isin 119869 = [0 119879] (28)

Proof FromTheorem 12 we obtain

119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (29)

Hence using Definition 11 and the basic condition ofTheorem 13 it follows that

119909 (119905) lt 120598 forall119905 isin 119869 = [0 119879] (30)


Conflict of Interests

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

Acknowledgments

The authors sincerely thank the reviewers for their valuablesuggestions and useful comments that have led to the presentimproved version of the original manuscript This researchwas jointly supported by National Natural Science Foun-dation of China (no 11371027 and no 11071001) DoctoralFund ofMinistry of Education of China (no 20093401110001)and Major Program of Educational Commission of AnhuiProvince of China (no KJ2011A020)

References

[1] K Zhang andDQ Cao ldquoFurther results on asymptotic stabilityof linear neutral systems with multiple delaysrdquo Journal of theFranklin Institute Engineering and Applied Mathematics vol344 no 6 pp 858ndash866 2007

[2] D Q Cao and P He ldquoStability criteria of linear neutral systemswith a single delayrdquoAppliedMathematics and Computation vol148 no 1 pp 135ndash143 2004

[3] P T Nam and V N Phat ldquoAn improved stability criterion fora class of neutral differential equationsrdquo Applied MathematicsLetters vol 22 no 1 pp 31ndash35 2009

[4] P Balasubramaniam R Krishnasamy and R RakkiyappanldquoDelay-dependent stability of neutral systems with time-varying delays using delay-decomposition approachrdquo AppliedMathematical Modelling vol 36 no 5 pp 2253ndash2261 2012

[5] FWang ldquoExponential asymptotic stability for nonlinear neutralsystems with multiple delaysrdquo Nonlinear Analysis Real WorldApplications vol 8 no 1 pp 312ndash322 2007

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

[7] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[8] A Anatoly Kilbas andHM SrivastavaTheory andApplicationsof Fractional Differential Equations Elsevier Amsterdam TheNetherlands 2006

[9] T Kaczorek Selected Problems of Fractional Systems TheorySpringer Berlin Germany 2011

[10] Y Zhou F Jiao and J Li ldquoExistence and uniqueness forfractional neutral differential equations with infinite delayrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 3249ndash3256 2009

[11] S Liu X Li W Jiang and X Zhou ldquoMittag-Leffler stability ofnonlinear fractional neutral singular systemsrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 10pp 3961ndash3966 2012

[12] X-F Zhou J Wei and L-G Hu ldquoControllability of a fractionallinear time-invariant neutral dynamical systemrdquo Applied Math-ematics Letters vol 26 no 4 pp 418ndash424 2013

[13] S Liu andW Jiang ldquoAsymptotic stability of nonlinear descriptorsystems with infinite delaysrdquo Annals of Differential Equationsvol 26 no 2 pp 174ndash180 2010

[14] Z Zhang and J Wei ldquoSome results of the degenerate fractionaldifferential system with delayrdquo Computers amp Mathematics withApplications vol 62 no 3 pp 1284ndash1291 2011

[15] M P Lazarevic ldquoFinite time stability analysis of 119875119863120572 fractionalcontrol of robotic time-delay systemsrdquo Mechanics ResearchCommunications vol 33 no 2 pp 269ndash279 2006

[16] C Bonnet and J R Partington ldquoAnalysis of fractional delaysystems of retarded and neutral typerdquo Automatica vol 38 no8 pp 1133ndash1138 2002

[17] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

[18] M P Lazarevic and A M Spasic ldquoFinite-time stability analysisof fractional order time-delay systems Gronwallrsquos approachrdquoMathematical andComputerModelling vol 49 no 3-4 pp 475ndash481 2009

[19] H Ye J Gao and Y Ding ldquoA generalized Gronwall inequalityand its application to a fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 328 no 2 pp1075ndash1081 2007

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Stochastic AnalysisInternational Journal of


Remark 2 (see [7]) (i) Γ(119911 + 1) = 119911Γ(119911) 119911 isin C and for 119899 isin

Z+ Γ(119899 + 1) = 119899(119899 minus 1) = 119899(ii) Γ(119911)Γ(119911 + (12)) = radic1205872

1minus2119911

Γ(2119911) 2119911 = 0 minus1 minus2 (ii) Γ(119899 + (12)) = radic120587Γ(2119899 + 1)2

2119899

Γ(119899 + 1) = radic120587(2119899)

22119899

119899

Definition 3 (see [7]) The fractional integral of order 120572 withthe lower limit zero for any function 119891(119905) isin 119862([0 +infin) 119877)119905 ge 0 is defined as

119868120572

119891 (119905) = limℎrarr0

119899ℎ=119905

ℎ120572

119899

sum

119903=0

[

120572

119903]119891 (119905 minus 119903ℎ)

=

1

Γ (120572)

int

119905

0

(119905 minus 120579)120572minus1

119891 (120579) 119889120579 120572 gt 0

(3)

where [ 120572119903] = 120572(120572 + 1) sdot sdot sdot (120572 + 119903 minus 1)119903 and Γ(sdot) is the gamma

function

Definition 4 (see [7]) The Riemann-Liouville derivative oforder 120572 with the lower limit zero for any function 119891(119905) isin

119862([0 +infin) 119877) 119905 ge 0 is defined as

119897

D120572

119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0


119891 (120579) 119889120579

119899 minus 1 lt 120572 lt 119899

(4)

Definition 5 (see [7]) The Caputo derivative of order 120572 forany function 119891(119905) isin 119862

119899

([0 +infin) 119877) 119905 ge 0 is defined as

119888D120572119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0


119891(119899)

(120579) 119889120579

= 119868119899minus120572

119891 (119905) 119899 minus 1 lt 120572 lt 119899

(5)

Remark 6 (see [7]) (i) If a function 119891(119905) isin 119862119899

([0 +infin) 119877)119905 ge 0 then 119888D120572119891(119905) =

119897D120572119891(119905) minus sum119899minus1

119896=0(119891(119896)

(0)Γ(119896 minus 120572 +

1))119905119896minus120572(ii) 119888D120572119891(119905) = 0 119891(119905) = 119862 and 119862 is a constant

Definition 7 (see [9]) The Mittag-Leffler function in twoparameters is defined as

119864120572120573

(119911) =

infin

sum

119896=0

(119911)119896

Γ (120572119896 + 120573)

119911 isin C (6)

where 120572 gt 0 120573 gt 0 and 119911 isin C

Remark 8 (see [9]) (i) For 120573 = 1 1198641205721(120582119911120572

) = 119864120572(120582119911120572

) =

suminfin

119896=0(120582119896

(119911120572

)119896

Γ(120572119896 + 1)) and 119864(11)

(119911) = 119890119911 119911 isin C

(ii) For120573 = 1 thematrix extension of the aforementionedMittag-Leffler function has the following representation119864120572(119860119905120572

) = suminfin

119896=0((119860119896

(119905120572

)119896

Γ(120572119896 + 1)) 119911 isin C and119888D120572119864

120572(119860119905120572

) = 119860119864120572(119860119905120572

)

Lemma 9 (see [19] generalized Gronwallrsquos inequality) Sup-pose 119909(119905) 119886(119905) are nonnegative and local integrable on 0 le 119905 lt

119879 some 119879 le infin and 119892(119905) is a nonnegative nondecreasing

continuous function defined on 0 le 119905 lt 119879 119892(119905) le 119872 =

constant 120572 gt 0 with

119909 (119905) le 119886 (119905) + 119892 (119905) int

119905

0

(119905 minus 119904)120572minus1

119909 (119904) 119889119904 (7)

on this interval Then

119909 (119905) le 119886 (119905) + 119892 (119905) int

119905

0

[

infin

sum

119899=1

(119892 (119905) Γ (120572))119899

Γ (119899120572)

(119905 minus 119904)119899120572minus1

119886 (119904)] 119889119904

0 le 119905 lt 119879

(8)

Lemma 10 (see [19]) Under the hypothesis of Theorem 13 let119886(119905) be a nondecreasing function on [0 119879) Then

119909 (119905) le 119886 (119905) 119864120572(119892 (119905) Γ (120572) 119905

120572

) (9)

where 119864120572is the Mittag-Leffler function

3 Main Results

In this section we discuss some problems of the neutralfractional time-delay system (1)

Let us denote by 119862([119886 119887]) the space of all continuous realfunctions defined on [119886 119887] and by 119862([119886 119887] 119877

119899

) the Banachspace of continuous functions mapping the interval [119886 119887]into 119877

119899 with the topology of uniform convergence Let 119862 =

119862([minus120591 0] 119877119899

) [119886 119887] = [minus120591 0] and designate the norm of anelement 120593 in 119862 by

10038171003817100381710038171205931003817100381710038171003817= supminus120591le119905le0

1003817100381710038171003817120593 (119905)

1003817100381710038171003817 (10)

Let 119883 = 119862([minus120591 119879] 119877119899

) and 119909(119905) = 120593(119905) 119905 isin [minus120591 0] beequipped with the norm

119909 (119905) = sup0le119905le119879

119909 (119905)

1003817100381710038171003817119909119905

1003817100381710038171003817= 119909 (119905 + 120579) = sup

minus120591le120579le0

119909 (119905 + 120579)

forall119909 isin 119883

(11)

where obviously 119909(119905) le 119909119905

Let ]max(sdot) be the largest singular value of matrix (sdot)namely

]119860119861max = ]max (119860) + ]max (119861) (12)

For convenience we denote ]max(119860) by ]0 ]max(119861) by ]

1

]max(119862) by ]2 ]max(119863) by ]

3 and ]max(119860119861) by ]

01 respec-

tively

Definition 11 (see [18]) The system given by (1) and satisfyinginitial condition 119909(119905) = 120593(119905) for 119905 isin [minus120591 0] is finite stablewith respect to 119905

0 120575 120598 119869 120575 lt 120598 119869 = [119905

0 1199050+ 119879] if and only if

10038171003817100381710038171205931003817100381710038171003817lt 120575 (13)

implies

119909 (119905) lt 120598 forall119905 isin 119869 (14)



119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) forall119905 isin 119869 = [0 119879]

(15)




+

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

(119860119909 (119904) + 119861119909 (119904 minus 120591)) 119889119904

= 120593 (0) + 119862120593 (minus120591) + 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

(119860119909 (119904) + 119861119909 (119904 minus 120591)) 119889119904 119905 ge 0

(16)


119909 (119905) le1003817100381710038171003817120593 (0)

1003817100381710038171003817+1003817100381710038171003817119862120593 (minus120591)

1003817100381710038171003817+ 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860119909 (119904) + 119861119909 (119904 minus 120591) 119889119904

le (1 + 119862)10038171003817100381710038171205931003817100381710038171003817+ 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860119909 (119904) + 119861119909 (119904 minus 120591) 119889119904

le (1 + 119862)10038171003817100381710038171205931003817100381710038171003817+ 119862 119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860 119909 (119904) 119889119904

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119861 119909 (119904 minus 120591) 119889119904

le (1 + ]2)10038171003817100381710038171205931003817100381710038171003817+ ]2119909 (119905 minus 120591)

+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904) 119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904 minus 120591) 119889119904 119905 ge 0

(17)


119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904) 119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904 minus 120591) 119889119904 0 le 119905 le 120591

(18)



0Γ(120572)) int

119905

0

|119905minus 119904|120572minus1

119909(119904)119889119904 and

(]1Γ(120572)) int

119905

0

|119905minus119904|120572minus1


1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817+

]01

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

0 le 119905 le 120591

(19)


119886 (119905) = (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817 (20)



1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]01119905120572

) 0 le 119905 le 120591 (21)


1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)

100381710038171003817100381710038171199091205910

10038171003817100381710038171003817119864120572(]01(119905 minus 1205910)120572

)

1205910le 119905 le 120591

0+ 120591 120591

0ge 0

(22)




120572(]120591120572) gt 119864

120572(]01120591120572

)

and 119864120572(](119905 minus 120591)

120572

)119864120572(]01120591120572

)119864120572(]119905120572) lt 1(1 + 2]

2)


1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) 0 le 119905 le 119899120591 le 119879

(23)



1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)1003817100381710038171003817119909119905minus120591

1003817100381710038171003817119864120572(]01120591120572

) (24)


1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(](119905 minus 120591)

120572

) (25)


1003817100381710038171003817le (1 + 2]

2) [(1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(](119905 minus 120591)

120572

)] 119864120572(]01120591120572

)

= (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572)

times

(1 + 2]2) 119864120572(](119905 minus 120591)

120572

) 119864120572(]01120591120572

)

119864120572(]119905120572)

le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572)

(26)


That is

119909 (119905) le1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (27)



(1 + 2]2) 119864120572(]119905120572) le

120598

120575

forall119905 isin 119869 = [0 119879] (28)


119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (29)


119909 (119905) lt 120598 forall119905 isin 119869 = [0 119879] (30)




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) forall119905 isin 119869 = [0 119879]

(15)




+

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

(119860119909 (119904) + 119861119909 (119904 minus 120591)) 119889119904

= 120593 (0) + 119862120593 (minus120591) + 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

(119905 minus 119904)120572minus1

(119860119909 (119904) + 119861119909 (119904 minus 120591)) 119889119904 119905 ge 0

(16)


119909 (119905) le1003817100381710038171003817120593 (0)

1003817100381710038171003817+1003817100381710038171003817119862120593 (minus120591)

1003817100381710038171003817+ 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860119909 (119904) + 119861119909 (119904 minus 120591) 119889119904

le (1 + 119862)10038171003817100381710038171205931003817100381710038171003817+ 119862119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860119909 (119904) + 119861119909 (119904 minus 120591) 119889119904

le (1 + 119862)10038171003817100381710038171205931003817100381710038171003817+ 119862 119909 (119905 minus 120591)

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119860 119909 (119904) 119889119904

+

1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119861 119909 (119904 minus 120591) 119889119904

le (1 + ]2)10038171003817100381710038171205931003817100381710038171003817+ ]2119909 (119905 minus 120591)

+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904) 119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904 minus 120591) 119889119904 119905 ge 0

(17)


119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904) 119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1

119909 (119904 minus 120591) 119889119904 0 le 119905 le 120591

(18)



0Γ(120572)) int

119905

0

|119905minus 119904|120572minus1

119909(119904)119889119904 and

(]1Γ(120572)) int

119905

0

|119905minus119904|120572minus1


1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817+

]0

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

+

]1

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817+

]01

Γ (120572)

int

119905

0

|119905 minus 119904|120572minus1 1003817

100381710038171003817119909119904

1003817100381710038171003817119889119904

0 le 119905 le 120591

(19)


119886 (119905) = (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817 (20)



1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]01119905120572

) 0 le 119905 le 120591 (21)


1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)

100381710038171003817100381710038171199091205910

10038171003817100381710038171003817119864120572(]01(119905 minus 1205910)120572

)

1205910le 119905 le 120591

0+ 120591 120591

0ge 0

(22)




120572(]120591120572) gt 119864

120572(]01120591120572

)

and 119864120572(](119905 minus 120591)

120572

)119864120572(]01120591120572

)119864120572(]119905120572) lt 1(1 + 2]

2)


1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) 0 le 119905 le 119899120591 le 119879

(23)



1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)1003817100381710038171003817119909119905minus120591

1003817100381710038171003817119864120572(]01120591120572

) (24)


1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(](119905 minus 120591)

120572

) (25)


1003817100381710038171003817le (1 + 2]

2) [(1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(](119905 minus 120591)

120572

)] 119864120572(]01120591120572

)

= (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572)

times

(1 + 2]2) 119864120572(](119905 minus 120591)

120572

) 119864120572(]01120591120572

)

119864120572(]119905120572)

le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572)

(26)


That is

119909 (119905) le1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (27)



(1 + 2]2) 119864120572(]119905120572) le

120598

120575

forall119905 isin 119869 = [0 119879] (28)


119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (29)


119909 (119905) lt 120598 forall119905 isin 119869 = [0 119879] (30)




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










That is

119909 (119905) le1003817100381710038171003817119909119905

1003817100381710038171003817le (1 + 2]

2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (27)



(1 + 2]2) 119864120572(]119905120572) le

120598

120575

forall119905 isin 119869 = [0 119879] (28)


119909 (119905) le (1 + 2]2)10038171003817100381710038171205931003817100381710038171003817119864120572(]119905120572) (29)


119909 (119905) lt 120598 forall119905 isin 119869 = [0 119879] (30)




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Finite-Time Stability of Neutral Fractional Time …downloads.hindawi.com/journals/aaa/2014/610547.pdf · Finite-Time Stability of Neutral Fractional Time-Delay Systems