Research ArticleSufficient Conditions on the Exponential Stability of NeutralStochastic Differential Equations with Time-Varying Delays

Yanwei Tian1 and Baofeng Chen2

1 School of Mathematics and Statistics Anyang Normal University Anyang Henan 455002 China2 School of Mathematics and Physics Anyang Institute of Technology Anyang Henan 455002 China

Correspondence should be addressed to Yanwei Tian tyw994126com

Received 24 January 2014 Revised 26 April 2014 Accepted 26 April 2014 Published 12 May 2014

Academic Editor Shuping He

Copyright copy 2014 Y Tian and B ChenThis 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

The exponential stability is investigated for neutral stochastic differential equations with time-varying delays Based on theLyapunov stability theory and linear matrix inequalities (LMIs) technique some delay-dependent criteria are established toguarantee the exponential stability in almost sure sense Finally a numerical example is provided to illustrate the feasibility ofthe result

1 Introduction

Neutral differential equations are well-known models frommany areas of science and engineering wherein quiteoften the future state of such systems depends not only onthe present state but also involves derivatives with delaysDeterministic neutral differential equations were originallyintroduced by Hale and Meyer [1] and discussed in Hale andLunel (see [2]) and Kolmanovskii et al (for details see also[3 4]) among others Motivated by chemical engineeringsystems as well as theory of aeroelasticity stochastic neutraldelay systems have been intensively studied over recent year[5ndash9] Mao initiated the study of exponential stability ofneutral stochastic differential delay equations in [5] while [9]incorporated Razumikhinis approach in neutral stochasticfunctional differential equations to investigate the stabilityproblem It is pointed out in Section 5 [10] that the conditionsimposed in [5 9] make the theory not applicable to the delayequation

More recently Luo et al [6] proposed new criteria onexponential stability of neutral stochastic delay differentialequations In [11 12] Milosevic investigated the almostsure exponential stability of a class of highly nonlinear

neutral stochastic differential equations with time-dependentdelay and some sufficient conditions were given for theconsidered systems However when the exponential stabilityof the neutral system with time-delay is considered onealways assumes that the derivative of the delay functionis less than 1 (eg [6]) Meanwhile the delay-independentconditions in [6 10] are restricted when the delay issmall On the other hand some results are proposed onstochastic Markovian jumping systems (eg [13ndash20]) andfinite-time problems of stochastic systems (eg [18ndash22])which can provide some useful methods and techniquesfor the neutral stochastic systems This paper aims todevelop the exponential stability in almost sure sense of theneutral stochastic differential equations with time-varyingdelays Under the weaker assumptions that the derivativeof time delay is less than some constant sufficient con-ditions for the exponential stability are given in terms oflinear matrix inequality (LMI) based on Lyapunov stabilitytheory which can be checked easily by MATLAB LMIToolbox

The paper is organized as follows In the remainder ofthis section we recall some preliminaries mainly from [5] InSection 3 we state the main results on exponential stability

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 391461 6 pageshttpdxdoiorg1011552014391461

2 Abstract and Applied Analysis

Section 4 will provide numerical examples to illustrate thefeasibility and effectiveness of the results and the conclusionwill be made in Section 5

2 Preliminaries

Throughout this paper unless otherwise specified let ΩF 119875 be a complete probability space with a filtrationF119905119905ge0

satisfying the usual conditions (ie right con-tinuous and 119865

0containing all 119875-null sets) Let 119908(119905) =

(1199081(119905) 1199082(119905) 119908

119898(119905))119899 be 119898-dimensional Wiener process

defined on the probability space Let | sdot | denote the Euclideannorm in R119899 119860119879 stands for the transpose of the vector ormatrix 119860 If 119860 is a matrix its trace norm is denoted by|119860| = radic119860119879119860 119886 or 119887 denotes max119886 119887 120582max(sdot) 120582min(sdot) aremaximum eigenvalue and minimum eigenvalue respective-ly

Consider the following 119899-dimensional neutral stochasticdifferential delay equations with time-varying delays

d [119909 (119905) minus 119866 (119909 (119905 minus 120575 (119905)))]

= 119891 (119909 (119905) 119909 (119905 minus 120575 (119905)) 119905) d119905

+ 119892 (119909 (119905) 119909 (119905 minus 120575 (119905)) 119905) d119908 (119905)

119909 (119905) = 120585 (119905) isin 119862119887

F0([minus120591 0] R

119899) 119905 isin [minus120591 0]

(1)

where 119891 R119899 times R119899 times R+

rarr R119899 119892 R119899 times R119899 times

R+

rarr R119899times119898 and 119866 isin 119862(R119899R119899) The functions 120575(119905)

R+

rarr [0 120591] are continuously differentiable such that 0 le

120575(119905) le 120575 120575(119905) le 120575 Let 119862([minus120591 0]R119899) denote the family ofcontinuous functions 120601 from [minus120591 0] to R119899 with the norm120601 = sup

minus120591le120579le0|120601(120579)| Let 119862119887F0([minus120591 0]R

119899) be the family

of allF0-measurable119862([minus120591 0]R119899)-valued random variables

120585 = 120585(120579) 120591 le 120579 le 0 such that supminus120591le120579le0

119864|120585(120579)|2lt infin To

guarantee the existence and uniqueness of the solution wefirst list the following hypotheses

(H1) Both the functionals 119891 and 119892 satisfy the uni-form Lipschitz conditions That is there is a diago-nal positive matrix 119871 = diag 119871

1 1198712 119871

119899 such

that1003816100381610038161003816119891 (119909 119910 119905) minus 119891 (119909 119910 119905)

1003816100381610038161003816 or1003816100381610038161003816119892 (119909 119910 119905) minus 119892 (119909 119910 119905)

1003816100381610038161003816

le |119871 (119909 minus 119909)| +1003816100381610038161003816119871 (119910 minus 119910)

1003816100381610038161003816

(2)

for all 119905 ge 0 and those 119909 119910 119909 119910 isin R119899

(H2) There is a constant 119896 isin (0 1) such that for all 120601

1 1206012isin

119862119887

F0([minus120591 0]R119899)

1003816100381610038161003816119866 (1206011) minus 119866 (120601

2)10038161003816100381610038162

le 119896 supminus120591le120579le0

10038161003816100381610038161206011 (120579) minus 1206012(120579)

10038161003816100381610038162

(3)

It is well known (see eg [3]) that under hypotheses H1 H2

(1) has a unique continuous solution on 119905 ge minus120591To obtain sufficient conditions on almost sure expo-

nential stability the following lemmas and definition aregiven

Lemma 1 (see [23]) For any positive definite constant matrix119872 isin R119899times119899 scalar 119903 gt 0 and vector function 119891(sdot) [0 119903] rarr

R119899 such that the integrations in the following are well definedthen the following inequality holds

(int119903

0

119891(119904)119889119904)

119879

119872(int119903

0

119891 (119904) 119889119904) le 119903int119903

0

119891119879(119904)119872119891 (119904) 119889119904 (4)

The following semimartingale convergence theoremwill play animportant role in the later parts

Lemma 2 (see [24]) Let 119860(119905) and 119880(119905) be two continuousadapted increasing processed on 119905 ge 0 with 119860(0) = 119880(0) = 0

as Let119872(119905) be a real-valued continuous local martingale with119872(0) = 0 as Let 120589 be a nonnegative F

0-measurable random

variable Define

119883(119905) = 120589 + 119860 (119905) minus 119880 (119905) + 119872 (119905) 119891119900119903 119905 ge 0 (5)

If119883(119905) is nonnegative then

lim119905rarrinfin

119860 (119905) lt infin sub lim119905rarrinfin

119883 (119905) lt infin

cap lim119905rarrinfin

119880 (119905) lt infin 119886119904

(6)

where 119861 sub 119863 as means 119875(119861 cap 119863) = 0 In particular iflim119905rarrinfin

119860(119905) lt infin as then for almost all 119908 isin Ω

lim119905rarrinfin

119883(119905) lt infin lim119905rarrinfin

119880 (119905) lt infin (7)

that is both119883(119905) and119880(119905) converge to finite random variables

Definition 3 (see [25]) The equilibrium of solution 119909(119905) 119905 ge

0 of (1) is said to be almost sure exponentially stable if thereexists a constant 120576 gt 0 such that

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus120576 as (8)

for any bounded initial condition 120585

3 Main Results

Theorem 4 Let hypotheses H1 H2hold System (1) is almost

sure exponentially stable if there exists positive definite matrixsuch that the following LMI holds

Abstract and Applied Analysis 3

Ω =

(((((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

50

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

50

lowast lowast Ξ33

0 minus119873119879

3119880

lowast lowast lowast Ξ44

minus119873119879

40

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

50

lowast lowast lowast lowast lowast minus120576119868

)))))))))))

)

lt 0 (9)

where

Ξ11

= 120573119875 + 119890120573120575119876 + 119875 + 2119873

119879

1

Ξ22

= (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575))119876 minus 21198732

Ξ33

= minus 2119880 + 119890120573120575119878 + 2120582max (119875) 119871

119879119871

+2

120573120582max (119877) (119890

120573120575minus 1) 119871

1

Ξ44

= minus (1 minus 120575) 119878 + 2120582max (119875) 119871119879119871

+2

120573120582max (119877) (119890

120573120575minus 1) 119871

1+ 120576119896119868

(10)

Proof To confirm that the stochastic neutral differentialequation (1) is mean-square exponentially stable with decayrate 120573 we define a Lyapunov-Krasovskii functional119881(119909(119905) 119905)as follows

119881 (119909 (119905) 119905) = 119890120573119905120588119879(119905) 119875120588 (119905)

+ int119905

119905minus120575(119905)

119890120573(119904+120575)

120588119879(119904) 119876120588 (119904) d119904

+ int0

minus120575

int119905

119905+120579

119890120573(119904minus120579)

119891119879(119909 (119904) 119910 (119904) 119904)

times 119877119891 (119909 (119904) 119910 (119904) 119904) d119904d120579

+ int119905

119905minus120575(119905)

119890120573(119904+120575)

119909119879(119904) 119878119909 (119904) d119904

(11)

For simplicity let 119910(119905) = 119909(119905 minus 120575(119905)) 120588(119905) = 119909(119905) minus 119866(119910(119905)) Bygeneralizing Itorsquos formula we have that

119864119881 (120588 (119905) 119905) = 119864119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

(12)

Then the derivative of119881(120588(119905) 119905) along the solution of (1) gives

119871119881 (119909 (119905) 119910 (119905) 119905)

= 120573119890120573119905120588119879(119905) 119875120588 (119905) + 2119890

120573119905120588(119905)119879119875119891 (119909 (119905) 119910 (119905) 119905)

+ 119890120573119905119892119879(119909 (119905) 119910 (119905) 119905) 119875119892 (119909 (119905) 119910 (119905) 119905)

+ 119890120573(119905+120575)

120588 (119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905 minus 120575 (119905)) 119876120588 (119905 minus 120575 (119905)) (1 minus 120575 (119905))

+1

120573119890120573119905(119890120573120575

minus 1)119891119879(119909 (119905) 119910 (119905) 119905) 119877119891 (119909 (119905) 119910 (119905) 119905)

minus 119890120573119905int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573(119905+120575)

119909 (119905) 119878119909 (119905) minus 119890120573(119905minus120575(119905))+120575

times 119909119879(119905 minus 120575 (119905)) 119878119909 (119905 minus 120575 (119905)) (1 minus 120575 (119905))

(13)

Note that from system (1) and Newton-Leibniz formula wehave

119872 = (120588 (119905) minus 120588 (119905 minus 120575 (119905)) minus int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904

minusint119905

119905minus120575(119905)

119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)) = 0

(14)

By calculation it is clear that

119891119879(119909 (119905) 119910 (119905) 119905) 119877119891 (119909 (119905) 119910 (119905) 119905)

le 120582max (119877) 119891119879(119909 (119905) 119910 (119905) 119905) 119891 (119909 (119905) 119910 (119905) 119905)

le 120582max (119877)1003816100381610038161003816119891 (119909 (119905) 119910 (119905) 119905)

10038161003816100381610038162

le 2120582max (119877) (119909119879(119905) 119871119879119871119909 (119905) + 119909

119879(119905 minus 120575 (119905)) 119871

119879119871119910 (119905))

(15)

4 Abstract and Applied Analysis

and then by which we have

2120588119879(119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 119891119879(119909 (119905) 119910 (119905) 119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) (119909119879(119905) 119871119879119871119909 (119905) + 119910

119879(119905) 119871119879119871119910 (119905))

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905) 119876120588 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905120588119879(119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

119909119879(119905) 119878119909 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905119909119879(119905) 119878119909 (119905)

(16)

Moreover by Lemma 2 one can get

minus int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minusint119905

119905minus120575(119905)

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(17)

Letting 1198711= 119871119879119871 and substituting (14)ndash(17) into (13) yield

119871119881 (119909 (119905) 119910 (119905) 119905)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) 119909119879(119905) 1198711119909 (119905)

+ 2120582max (119875) 119910119879(119905) 1198711119910 (119905)

+ (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575)) 120588119879(119905) 119876120588 (119905)

+ 119890120573120575120588119879(119905) 1198761120588 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(18)

Furthermore from (14) it follows that

119860 = 2120578119879(119873119879

1 119873119879

2 119873119879

3 119873119879

4 119873119879

5)119879

119872 = 0

119861 = 2119909119879(119905) 119880 [120588 (119905) minus 119909 (119905) + 119866 (119910 (119905))] = 0

(19)

where 120578 = (120588119879 120588119879(119905 minus 120575(119905)) 119909119879(119905) 119910119879(119905) (int119905

119905minus120575(119905)119891(119909 119910 119904)

d119904)119879)119879 and 119873119894(1 le 119894 le 5) 119880 are matrices with compatible

dimensionsIt can be shown that

int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573119905(119860 + 119861) + 119890

1205731199051198721198791198751(119872(119905) + int

119905

119905minus120575(119905)

119892 (119909 119910 119904) d119904)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 119909119879(119905) 1198711119909 (119905) + 119890

120573120575120588119879(119905) 119876120588 (119905)

+ 119910119879(119905) 1198711119910 (119905) + 2120582max (119875) 119909

119879(119905) 1198711119909 (119905)

+ 2120582max119910119879(119905) 1198711119910 (119905)

minus (1 minus 120575) 120588119879(119905) 119876120588 (119905) + 119890

120573120575119909119879(119905) 119878119909 (119905)

minus (1 minus 120575) 119909119879(119905) 119878119909 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

le 119890120573119905120578119879Ω120578 + 120576

minus1119909119879(119905) 119880119880

119879119909 (119905)

(20)

whereΩ is defined as

Ω =

((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

5

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

5

lowast lowast Ξ33

0 minus119873119879

3

lowast lowast lowast Ξ44

minus119873119879

4

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

5

))))))))

)

(21)

Abstract and Applied Analysis 5

By Schur complement we know that 120578119879Ω120578 +

120576minus1119909119879(119905)119880119880

119879119909(119905) lt 0 On the other hand it follows

that

119881 (120588 (119905) 119905) = 119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ int119905

0

2119890120573119904119909119904(119905) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(22)

Note that 120585 is bounded and119881119866 are continuous then119881(120588(0))must be nonnegative bounded Moreover L119881(119909 119910 119905) le 0

can be obtained directly

119881 (120588 (119905) 119905) le 119881 (120588 (0) 0)

+ int119905

0

2119890120573119905119909119879(119904) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(23)

By applying Lemma 2 to (23) one sees that

lim119905rarrinfin

sup119881 (120588 (119905) 119905) lt infin (24)

hence there exists a positive random variable 120577 satisfying

sup0le119905ltinfin

1198901205731199051003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))

10038161003816100381610038162

le 120577 (25)

Since for any 1205763isin (0 1)

1003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))10038161003816100381610038162

ge (1 minus 120576minus1

3) |119909 (119905)|

2minus (1205763minus 1)

1003816100381610038161003816119866 (119910 (119905))10038161003816100381610038162

(26)

we must have

sup0le119905le119879

119890120573119905|119909 (119905)|

2

le 120577 +1198962

1205763

sup0le119905le119879

119890120573119905 1003816100381610038161003816119910 (119905)

10038161003816100381610038162

le 120577 + 11989621198901205731205911003817100381710038171003817120585

10038171003817100381710038172

+1198962

1205763

119890120573120591 sup0le119905le119879

119890120573119905|119909 (119905)|

2

(27)

From the above inequality (26) it yields the desired result

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus

120573

2 (28)

That completes the proof

4 Example

In this section a numerical example will be given to illustratethat the proposed method is effective

Example 1 Consider the following system

119889 [1199091(119905) minus 01119909

2(119905 minus 120575 (119905))] = minus 119909

1(119905) 1199092(119905 minus 120575 (119905)) d119905

+ 1199091(119905) sin2 (119905 minus 120575 (119905)) d120596 (119905)

119889 [1199092(119905) minus 01119909

1(119905 minus 120575 (119905))] = minus 119909

2(119905) 1199091(119905 minus 120575 (119905)) d119905

+ 1199092(119905) cos2 (119905 minus 120575 (119905)) d120596 (119905)

(29)

where the delay function is defined as 120575(119905) = (14) sin(119905) 119905 gt0 It is obvious that (29) satisfies the assumptions H

1and H

2

and here 119871 = 119868 119896 = 01 Moreover since 120575 = (14) cos(119905)then 120575 = 14

According to Theorem 4 and employing MATLAB LMIToolbox it is relatively easy to deduce that the neutralstochastic differential equation (29) is almost sure exponen-tially stable

Remark 5 Comparing with some existing sufficient criteriafor neural stochastic differential equations (eg [6 11 12])the obtained result is given in terms of linear matrix inequal-ity (LMI) which can be easily checked by MATLAB LMIToolbox

5 Conclusion

The exponential stability is investigated for a class of neutralstochastic differential equations with time-varying delaysIn order to overcome the difficulties we introduce suitableLyapunov functionals and employ linear matrix inequalities(LMIs) technique and then a delay-dependent criteria aregiven to check the almost sure exponential stability of theconcerned equations

Conflict of Interests

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

References

[1] J K Hale and K R Meyer A Class of Functional Equations ofNeutral Type Memoirs of the American Mathematical Societyno 76 American Mathematical Society Providence RI USA1967

[2] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[3] V B Kolmanovskii and V R Nosov Stability of Functional-Differential Equations vol 180 of Mathematics in Science andEngineering Academic Press New York NY USA 1986

[4] V B Kolmanovskii and A Myshkis Applied Theory of Func-tional Differential Equations Kluwer Academic PublishersNorwell Mass USA 1992

[5] X Mao ldquoExponential stability in mean square of neutralstochastic differential-functional equationsrdquo Systems amp ControlLetters vol 26 no 4 pp 245ndash251 1995

[6] Q Luo X Mao and Y Shen ldquoNew criteria on exponentialstability of neutral stochastic differential delay equationsrdquoSystems amp Control Letters vol 55 no 10 pp 826ndash834 2006

[7] V B Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999

[8] M SMahmoud ldquoRobust119867infincontrol of linear neutral systemsrdquo

Automatica vol 36 no 5 pp 757ndash764 2000[9] X Mao ldquoRazumikhin-type theorems on exponential stability

of neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

6 Abstract and Applied Analysis

[10] XMao ldquoAsymptotic properties of neutral stochastic differentialdelay equationsrdquo Stochastics and Stochastics Reports vol 68 no3-4 pp 273ndash295 2000

[11] M Milosevic ldquoHighly nonlinear neutral stochastic differentialequations with time-dependent delay and the Euler-MaruyamamethodrdquoMathematical and Computer Modelling vol 54 no 9-10 pp 2235ndash2251 2011

[12] M Milosevic ldquoAlmost sure exponential stability of solutions tohighly nonlinear neutral stochastic differential equations withtime-dependent delay and the Euler-Maruyama approxima-tionrdquo Mathematical and Computer Modelling vol 57 no 3-4pp 887ndash899 2013

[13] S He ldquoResilient 1198712-119871infinfiltering of uncertain Markovian jump-

ing systemswithin the finite-time intervalrdquoAbstract andAppliedAnalysis vol 2013 Article ID 791296 7 pages 2013

[14] S He and F Liu ldquoAdaptive observer-based fault estimation forstochastic Markovian jumping systemsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 176419 11 pages 2012

[15] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198712minus119871infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[16] H Shen Y Chu S Xu and Z Zhang ldquoDelay-dependent119867infin

control for jumping delayed systems with two markovprocessesrdquo International Journal of Control Automation andSystems vol 9 no 3 pp 437ndash441 2011

[17] H Shen J Park L Zhang and Z Wu ldquoRobust extendeddissipative control for sampled-data Markov jump systemsrdquoInternational Journal of Control 2014

[18] J ChengH Zhu S Zhong Y Zhang andY Li ldquoFinite-time119867infin

control for a class of discrete-time Markov jump systems withpartly unknown time-varying transition probabilities subject toaverage dwell time switchingrdquo Journal of Systems Science 2013

[19] J Cheng H Zhu S Zhong Y Zeng and L Hou ldquoFinite-time 119867

infinfiltering for a class of discrete-time Markovian jump

systems with partly unknown transition probabilitiesrdquo Interna-tional Journal of Adaptive Control and Signal Processing 2013

[20] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 pp 768ndash774 2013

[21] S He and F Liu ldquoFinite-time 119867infin

control of nonlinear jumpsystems with timedelays via dynamic observer-based statefeedbackrdquo IEEE Transactions on Fuzzy Systems vol 20 no 4pp 605ndash614 2012

[22] S-P He and F Liu ldquoRobust finite-time119867infincontrol of stochastic

jump systemsrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1336ndash1341 2010

[23] R Sh Liptser and A N Shiryayev Theory of Martingales vol49 of Mathematics and Its Applications (Soviet Series) KluwerAcademic Publishers Dordrecht The Netherlands 1989

[24] K Gu ldquoAn integral inequality in the stability problem oftime-delay systemsrdquo in Proceedings of the 39th IEEE Confernceon Decision and Control pp 2805ndash2810 Sydney AustraliaDecember 2000

[25] Q Luo and Y Zhang ldquoAlmost sure exponential stability ofstochastic reaction diffusion systemsrdquo Nonlinear Analysis The-ory Methods ampApplications vol 71 no 12 pp e487ndashe493 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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

2 Abstract and Applied Analysis

Section 4 will provide numerical examples to illustrate thefeasibility and effectiveness of the results and the conclusionwill be made in Section 5

2 Preliminaries

Throughout this paper unless otherwise specified let ΩF 119875 be a complete probability space with a filtrationF119905119905ge0

satisfying the usual conditions (ie right con-tinuous and 119865

0containing all 119875-null sets) Let 119908(119905) =

(1199081(119905) 1199082(119905) 119908

119898(119905))119899 be 119898-dimensional Wiener process

defined on the probability space Let | sdot | denote the Euclideannorm in R119899 119860119879 stands for the transpose of the vector ormatrix 119860 If 119860 is a matrix its trace norm is denoted by|119860| = radic119860119879119860 119886 or 119887 denotes max119886 119887 120582max(sdot) 120582min(sdot) aremaximum eigenvalue and minimum eigenvalue respective-ly

Consider the following 119899-dimensional neutral stochasticdifferential delay equations with time-varying delays

d [119909 (119905) minus 119866 (119909 (119905 minus 120575 (119905)))]

= 119891 (119909 (119905) 119909 (119905 minus 120575 (119905)) 119905) d119905

+ 119892 (119909 (119905) 119909 (119905 minus 120575 (119905)) 119905) d119908 (119905)

119909 (119905) = 120585 (119905) isin 119862119887

F0([minus120591 0] R

119899) 119905 isin [minus120591 0]

(1)

where 119891 R119899 times R119899 times R+

rarr R119899 119892 R119899 times R119899 times

R+

rarr R119899times119898 and 119866 isin 119862(R119899R119899) The functions 120575(119905)

R+

rarr [0 120591] are continuously differentiable such that 0 le

120575(119905) le 120575 120575(119905) le 120575 Let 119862([minus120591 0]R119899) denote the family ofcontinuous functions 120601 from [minus120591 0] to R119899 with the norm120601 = sup

minus120591le120579le0|120601(120579)| Let 119862119887F0([minus120591 0]R

119899) be the family

of allF0-measurable119862([minus120591 0]R119899)-valued random variables

120585 = 120585(120579) 120591 le 120579 le 0 such that supminus120591le120579le0

119864|120585(120579)|2lt infin To

guarantee the existence and uniqueness of the solution wefirst list the following hypotheses

(H1) Both the functionals 119891 and 119892 satisfy the uni-form Lipschitz conditions That is there is a diago-nal positive matrix 119871 = diag 119871

1 1198712 119871

119899 such

that1003816100381610038161003816119891 (119909 119910 119905) minus 119891 (119909 119910 119905)

1003816100381610038161003816 or1003816100381610038161003816119892 (119909 119910 119905) minus 119892 (119909 119910 119905)

1003816100381610038161003816

le |119871 (119909 minus 119909)| +1003816100381610038161003816119871 (119910 minus 119910)

1003816100381610038161003816

(2)

for all 119905 ge 0 and those 119909 119910 119909 119910 isin R119899

(H2) There is a constant 119896 isin (0 1) such that for all 120601

1 1206012isin

119862119887

F0([minus120591 0]R119899)

1003816100381610038161003816119866 (1206011) minus 119866 (120601

2)10038161003816100381610038162

le 119896 supminus120591le120579le0

10038161003816100381610038161206011 (120579) minus 1206012(120579)

10038161003816100381610038162

(3)

It is well known (see eg [3]) that under hypotheses H1 H2

(1) has a unique continuous solution on 119905 ge minus120591To obtain sufficient conditions on almost sure expo-

nential stability the following lemmas and definition aregiven

Lemma 1 (see [23]) For any positive definite constant matrix119872 isin R119899times119899 scalar 119903 gt 0 and vector function 119891(sdot) [0 119903] rarr

R119899 such that the integrations in the following are well definedthen the following inequality holds

(int119903

0

119891(119904)119889119904)

119879

119872(int119903

0

119891 (119904) 119889119904) le 119903int119903

0

119891119879(119904)119872119891 (119904) 119889119904 (4)

The following semimartingale convergence theoremwill play animportant role in the later parts

Lemma 2 (see [24]) Let 119860(119905) and 119880(119905) be two continuousadapted increasing processed on 119905 ge 0 with 119860(0) = 119880(0) = 0

as Let119872(119905) be a real-valued continuous local martingale with119872(0) = 0 as Let 120589 be a nonnegative F

0-measurable random

variable Define

119883(119905) = 120589 + 119860 (119905) minus 119880 (119905) + 119872 (119905) 119891119900119903 119905 ge 0 (5)

If119883(119905) is nonnegative then

lim119905rarrinfin

119860 (119905) lt infin sub lim119905rarrinfin

119883 (119905) lt infin

cap lim119905rarrinfin

119880 (119905) lt infin 119886119904

(6)

where 119861 sub 119863 as means 119875(119861 cap 119863) = 0 In particular iflim119905rarrinfin

119860(119905) lt infin as then for almost all 119908 isin Ω

lim119905rarrinfin

119883(119905) lt infin lim119905rarrinfin

119880 (119905) lt infin (7)

that is both119883(119905) and119880(119905) converge to finite random variables

Definition 3 (see [25]) The equilibrium of solution 119909(119905) 119905 ge

0 of (1) is said to be almost sure exponentially stable if thereexists a constant 120576 gt 0 such that

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus120576 as (8)

for any bounded initial condition 120585

3 Main Results

Theorem 4 Let hypotheses H1 H2hold System (1) is almost

sure exponentially stable if there exists positive definite matrixsuch that the following LMI holds

Abstract and Applied Analysis 3

Ω =

(((((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

50

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

50

lowast lowast Ξ33

0 minus119873119879

3119880

lowast lowast lowast Ξ44

minus119873119879

40

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

50

lowast lowast lowast lowast lowast minus120576119868

)))))))))))

)

lt 0 (9)

where

Ξ11

= 120573119875 + 119890120573120575119876 + 119875 + 2119873

119879

1

Ξ22

= (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575))119876 minus 21198732

Ξ33

= minus 2119880 + 119890120573120575119878 + 2120582max (119875) 119871

119879119871

+2

120573120582max (119877) (119890

120573120575minus 1) 119871

1

Ξ44

= minus (1 minus 120575) 119878 + 2120582max (119875) 119871119879119871

+2

120573120582max (119877) (119890

120573120575minus 1) 119871

1+ 120576119896119868

(10)

Proof To confirm that the stochastic neutral differentialequation (1) is mean-square exponentially stable with decayrate 120573 we define a Lyapunov-Krasovskii functional119881(119909(119905) 119905)as follows

119881 (119909 (119905) 119905) = 119890120573119905120588119879(119905) 119875120588 (119905)

+ int119905

119905minus120575(119905)

119890120573(119904+120575)

120588119879(119904) 119876120588 (119904) d119904

+ int0

minus120575

int119905

119905+120579

119890120573(119904minus120579)

119891119879(119909 (119904) 119910 (119904) 119904)

times 119877119891 (119909 (119904) 119910 (119904) 119904) d119904d120579

+ int119905

119905minus120575(119905)

119890120573(119904+120575)

119909119879(119904) 119878119909 (119904) d119904

(11)

For simplicity let 119910(119905) = 119909(119905 minus 120575(119905)) 120588(119905) = 119909(119905) minus 119866(119910(119905)) Bygeneralizing Itorsquos formula we have that

119864119881 (120588 (119905) 119905) = 119864119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

(12)

Then the derivative of119881(120588(119905) 119905) along the solution of (1) gives

119871119881 (119909 (119905) 119910 (119905) 119905)

= 120573119890120573119905120588119879(119905) 119875120588 (119905) + 2119890

120573119905120588(119905)119879119875119891 (119909 (119905) 119910 (119905) 119905)

+ 119890120573119905119892119879(119909 (119905) 119910 (119905) 119905) 119875119892 (119909 (119905) 119910 (119905) 119905)

+ 119890120573(119905+120575)

120588 (119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905 minus 120575 (119905)) 119876120588 (119905 minus 120575 (119905)) (1 minus 120575 (119905))

+1

120573119890120573119905(119890120573120575

minus 1)119891119879(119909 (119905) 119910 (119905) 119905) 119877119891 (119909 (119905) 119910 (119905) 119905)

minus 119890120573119905int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573(119905+120575)

119909 (119905) 119878119909 (119905) minus 119890120573(119905minus120575(119905))+120575

times 119909119879(119905 minus 120575 (119905)) 119878119909 (119905 minus 120575 (119905)) (1 minus 120575 (119905))

(13)

Note that from system (1) and Newton-Leibniz formula wehave

119872 = (120588 (119905) minus 120588 (119905 minus 120575 (119905)) minus int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904

minusint119905

119905minus120575(119905)

119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)) = 0

(14)

By calculation it is clear that

119891119879(119909 (119905) 119910 (119905) 119905) 119877119891 (119909 (119905) 119910 (119905) 119905)

le 120582max (119877) 119891119879(119909 (119905) 119910 (119905) 119905) 119891 (119909 (119905) 119910 (119905) 119905)

le 120582max (119877)1003816100381610038161003816119891 (119909 (119905) 119910 (119905) 119905)

10038161003816100381610038162

le 2120582max (119877) (119909119879(119905) 119871119879119871119909 (119905) + 119909

119879(119905 minus 120575 (119905)) 119871

119879119871119910 (119905))

(15)

4 Abstract and Applied Analysis

and then by which we have

2120588119879(119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 119891119879(119909 (119905) 119910 (119905) 119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) (119909119879(119905) 119871119879119871119909 (119905) + 119910

119879(119905) 119871119879119871119910 (119905))

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905) 119876120588 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905120588119879(119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

119909119879(119905) 119878119909 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905119909119879(119905) 119878119909 (119905)

(16)

Moreover by Lemma 2 one can get

minus int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minusint119905

119905minus120575(119905)

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(17)

Letting 1198711= 119871119879119871 and substituting (14)ndash(17) into (13) yield

119871119881 (119909 (119905) 119910 (119905) 119905)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) 119909119879(119905) 1198711119909 (119905)

+ 2120582max (119875) 119910119879(119905) 1198711119910 (119905)

+ (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575)) 120588119879(119905) 119876120588 (119905)

+ 119890120573120575120588119879(119905) 1198761120588 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(18)

Furthermore from (14) it follows that

119860 = 2120578119879(119873119879

1 119873119879

2 119873119879

3 119873119879

4 119873119879

5)119879

119872 = 0

119861 = 2119909119879(119905) 119880 [120588 (119905) minus 119909 (119905) + 119866 (119910 (119905))] = 0

(19)

where 120578 = (120588119879 120588119879(119905 minus 120575(119905)) 119909119879(119905) 119910119879(119905) (int119905

119905minus120575(119905)119891(119909 119910 119904)

d119904)119879)119879 and 119873119894(1 le 119894 le 5) 119880 are matrices with compatible

dimensionsIt can be shown that

int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573119905(119860 + 119861) + 119890

1205731199051198721198791198751(119872(119905) + int

119905

119905minus120575(119905)

119892 (119909 119910 119904) d119904)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 119909119879(119905) 1198711119909 (119905) + 119890

120573120575120588119879(119905) 119876120588 (119905)

+ 119910119879(119905) 1198711119910 (119905) + 2120582max (119875) 119909

119879(119905) 1198711119909 (119905)

+ 2120582max119910119879(119905) 1198711119910 (119905)

minus (1 minus 120575) 120588119879(119905) 119876120588 (119905) + 119890

120573120575119909119879(119905) 119878119909 (119905)

minus (1 minus 120575) 119909119879(119905) 119878119909 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

le 119890120573119905120578119879Ω120578 + 120576

minus1119909119879(119905) 119880119880

119879119909 (119905)

(20)

whereΩ is defined as

Ω =

((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

5

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

5

lowast lowast Ξ33

0 minus119873119879

3

lowast lowast lowast Ξ44

minus119873119879

4

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

5

))))))))

)

(21)

Abstract and Applied Analysis 5

By Schur complement we know that 120578119879Ω120578 +

120576minus1119909119879(119905)119880119880

119879119909(119905) lt 0 On the other hand it follows

that

119881 (120588 (119905) 119905) = 119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ int119905

0

2119890120573119904119909119904(119905) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(22)

Note that 120585 is bounded and119881119866 are continuous then119881(120588(0))must be nonnegative bounded Moreover L119881(119909 119910 119905) le 0

can be obtained directly

119881 (120588 (119905) 119905) le 119881 (120588 (0) 0)

+ int119905

0

2119890120573119905119909119879(119904) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(23)

By applying Lemma 2 to (23) one sees that

lim119905rarrinfin

sup119881 (120588 (119905) 119905) lt infin (24)

hence there exists a positive random variable 120577 satisfying

sup0le119905ltinfin

1198901205731199051003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))

10038161003816100381610038162

le 120577 (25)

Since for any 1205763isin (0 1)

1003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))10038161003816100381610038162

ge (1 minus 120576minus1

3) |119909 (119905)|

2minus (1205763minus 1)

1003816100381610038161003816119866 (119910 (119905))10038161003816100381610038162

(26)

we must have

sup0le119905le119879

119890120573119905|119909 (119905)|

2

le 120577 +1198962

1205763

sup0le119905le119879

119890120573119905 1003816100381610038161003816119910 (119905)

10038161003816100381610038162

le 120577 + 11989621198901205731205911003817100381710038171003817120585

10038171003817100381710038172

+1198962

1205763

119890120573120591 sup0le119905le119879

119890120573119905|119909 (119905)|

2

(27)

From the above inequality (26) it yields the desired result

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus

120573

2 (28)

That completes the proof

4 Example

In this section a numerical example will be given to illustratethat the proposed method is effective

Example 1 Consider the following system

119889 [1199091(119905) minus 01119909

2(119905 minus 120575 (119905))] = minus 119909

1(119905) 1199092(119905 minus 120575 (119905)) d119905

+ 1199091(119905) sin2 (119905 minus 120575 (119905)) d120596 (119905)

119889 [1199092(119905) minus 01119909

1(119905 minus 120575 (119905))] = minus 119909

2(119905) 1199091(119905 minus 120575 (119905)) d119905

+ 1199092(119905) cos2 (119905 minus 120575 (119905)) d120596 (119905)

(29)

where the delay function is defined as 120575(119905) = (14) sin(119905) 119905 gt0 It is obvious that (29) satisfies the assumptions H

1and H

2

and here 119871 = 119868 119896 = 01 Moreover since 120575 = (14) cos(119905)then 120575 = 14

According to Theorem 4 and employing MATLAB LMIToolbox it is relatively easy to deduce that the neutralstochastic differential equation (29) is almost sure exponen-tially stable

Remark 5 Comparing with some existing sufficient criteriafor neural stochastic differential equations (eg [6 11 12])the obtained result is given in terms of linear matrix inequal-ity (LMI) which can be easily checked by MATLAB LMIToolbox

5 Conclusion

The exponential stability is investigated for a class of neutralstochastic differential equations with time-varying delaysIn order to overcome the difficulties we introduce suitableLyapunov functionals and employ linear matrix inequalities(LMIs) technique and then a delay-dependent criteria aregiven to check the almost sure exponential stability of theconcerned equations

Conflict of Interests

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

References

[1] J K Hale and K R Meyer A Class of Functional Equations ofNeutral Type Memoirs of the American Mathematical Societyno 76 American Mathematical Society Providence RI USA1967

[2] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[3] V B Kolmanovskii and V R Nosov Stability of Functional-Differential Equations vol 180 of Mathematics in Science andEngineering Academic Press New York NY USA 1986

[4] V B Kolmanovskii and A Myshkis Applied Theory of Func-tional Differential Equations Kluwer Academic PublishersNorwell Mass USA 1992

[5] X Mao ldquoExponential stability in mean square of neutralstochastic differential-functional equationsrdquo Systems amp ControlLetters vol 26 no 4 pp 245ndash251 1995

[6] Q Luo X Mao and Y Shen ldquoNew criteria on exponentialstability of neutral stochastic differential delay equationsrdquoSystems amp Control Letters vol 55 no 10 pp 826ndash834 2006

[7] V B Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999

[8] M SMahmoud ldquoRobust119867infincontrol of linear neutral systemsrdquo

Automatica vol 36 no 5 pp 757ndash764 2000[9] X Mao ldquoRazumikhin-type theorems on exponential stability

of neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

6 Abstract and Applied Analysis

[10] XMao ldquoAsymptotic properties of neutral stochastic differentialdelay equationsrdquo Stochastics and Stochastics Reports vol 68 no3-4 pp 273ndash295 2000

[11] M Milosevic ldquoHighly nonlinear neutral stochastic differentialequations with time-dependent delay and the Euler-MaruyamamethodrdquoMathematical and Computer Modelling vol 54 no 9-10 pp 2235ndash2251 2011

[12] M Milosevic ldquoAlmost sure exponential stability of solutions tohighly nonlinear neutral stochastic differential equations withtime-dependent delay and the Euler-Maruyama approxima-tionrdquo Mathematical and Computer Modelling vol 57 no 3-4pp 887ndash899 2013

[13] S He ldquoResilient 1198712-119871infinfiltering of uncertain Markovian jump-

ing systemswithin the finite-time intervalrdquoAbstract andAppliedAnalysis vol 2013 Article ID 791296 7 pages 2013

[14] S He and F Liu ldquoAdaptive observer-based fault estimation forstochastic Markovian jumping systemsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 176419 11 pages 2012

[15] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198712minus119871infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[16] H Shen Y Chu S Xu and Z Zhang ldquoDelay-dependent119867infin

control for jumping delayed systems with two markovprocessesrdquo International Journal of Control Automation andSystems vol 9 no 3 pp 437ndash441 2011

[17] H Shen J Park L Zhang and Z Wu ldquoRobust extendeddissipative control for sampled-data Markov jump systemsrdquoInternational Journal of Control 2014

[18] J ChengH Zhu S Zhong Y Zhang andY Li ldquoFinite-time119867infin

control for a class of discrete-time Markov jump systems withpartly unknown time-varying transition probabilities subject toaverage dwell time switchingrdquo Journal of Systems Science 2013

[19] J Cheng H Zhu S Zhong Y Zeng and L Hou ldquoFinite-time 119867

infinfiltering for a class of discrete-time Markovian jump

systems with partly unknown transition probabilitiesrdquo Interna-tional Journal of Adaptive Control and Signal Processing 2013

[20] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 pp 768ndash774 2013

[21] S He and F Liu ldquoFinite-time 119867infin

control of nonlinear jumpsystems with timedelays via dynamic observer-based statefeedbackrdquo IEEE Transactions on Fuzzy Systems vol 20 no 4pp 605ndash614 2012

[22] S-P He and F Liu ldquoRobust finite-time119867infincontrol of stochastic

jump systemsrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1336ndash1341 2010

[23] R Sh Liptser and A N Shiryayev Theory of Martingales vol49 of Mathematics and Its Applications (Soviet Series) KluwerAcademic Publishers Dordrecht The Netherlands 1989

[24] K Gu ldquoAn integral inequality in the stability problem oftime-delay systemsrdquo in Proceedings of the 39th IEEE Confernceon Decision and Control pp 2805ndash2810 Sydney AustraliaDecember 2000

[25] Q Luo and Y Zhang ldquoAlmost sure exponential stability ofstochastic reaction diffusion systemsrdquo Nonlinear Analysis The-ory Methods ampApplications vol 71 no 12 pp e487ndashe493 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Ω =

(((((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

50

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

50

lowast lowast Ξ33

0 minus119873119879

3119880

lowast lowast lowast Ξ44

minus119873119879

40

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

50

lowast lowast lowast lowast lowast minus120576119868

)))))))))))

)

lt 0 (9)

where

Ξ11

= 120573119875 + 119890120573120575119876 + 119875 + 2119873

119879

1

Ξ22

= (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575))119876 minus 21198732

Ξ33

= minus 2119880 + 119890120573120575119878 + 2120582max (119875) 119871

119879119871

+2

120573120582max (119877) (119890

120573120575minus 1) 119871

1

Ξ44

= minus (1 minus 120575) 119878 + 2120582max (119875) 119871119879119871

+2

120573120582max (119877) (119890

120573120575minus 1) 119871

1+ 120576119896119868

(10)

Proof To confirm that the stochastic neutral differentialequation (1) is mean-square exponentially stable with decayrate 120573 we define a Lyapunov-Krasovskii functional119881(119909(119905) 119905)as follows

119881 (119909 (119905) 119905) = 119890120573119905120588119879(119905) 119875120588 (119905)

+ int119905

119905minus120575(119905)

119890120573(119904+120575)

120588119879(119904) 119876120588 (119904) d119904

+ int0

minus120575

int119905

119905+120579

119890120573(119904minus120579)

119891119879(119909 (119904) 119910 (119904) 119904)

times 119877119891 (119909 (119904) 119910 (119904) 119904) d119904d120579

+ int119905

119905minus120575(119905)

119890120573(119904+120575)

119909119879(119904) 119878119909 (119904) d119904

(11)

For simplicity let 119910(119905) = 119909(119905 minus 120575(119905)) 120588(119905) = 119909(119905) minus 119866(119910(119905)) Bygeneralizing Itorsquos formula we have that

119864119881 (120588 (119905) 119905) = 119864119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

(12)

Then the derivative of119881(120588(119905) 119905) along the solution of (1) gives

119871119881 (119909 (119905) 119910 (119905) 119905)

= 120573119890120573119905120588119879(119905) 119875120588 (119905) + 2119890

120573119905120588(119905)119879119875119891 (119909 (119905) 119910 (119905) 119905)

+ 119890120573119905119892119879(119909 (119905) 119910 (119905) 119905) 119875119892 (119909 (119905) 119910 (119905) 119905)

+ 119890120573(119905+120575)

120588 (119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905 minus 120575 (119905)) 119876120588 (119905 minus 120575 (119905)) (1 minus 120575 (119905))

+1

120573119890120573119905(119890120573120575

minus 1)119891119879(119909 (119905) 119910 (119905) 119905) 119877119891 (119909 (119905) 119910 (119905) 119905)

minus 119890120573119905int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573(119905+120575)

119909 (119905) 119878119909 (119905) minus 119890120573(119905minus120575(119905))+120575

times 119909119879(119905 minus 120575 (119905)) 119878119909 (119905 minus 120575 (119905)) (1 minus 120575 (119905))

(13)

Note that from system (1) and Newton-Leibniz formula wehave

119872 = (120588 (119905) minus 120588 (119905 minus 120575 (119905)) minus int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904

minusint119905

119905minus120575(119905)

119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)) = 0

(14)

By calculation it is clear that

119891119879(119909 (119905) 119910 (119905) 119905) 119877119891 (119909 (119905) 119910 (119905) 119905)

le 120582max (119877) 119891119879(119909 (119905) 119910 (119905) 119905) 119891 (119909 (119905) 119910 (119905) 119905)

le 120582max (119877)1003816100381610038161003816119891 (119909 (119905) 119910 (119905) 119905)

10038161003816100381610038162

le 2120582max (119877) (119909119879(119905) 119871119879119871119909 (119905) + 119909

119879(119905 minus 120575 (119905)) 119871

119879119871119910 (119905))

(15)

4 Abstract and Applied Analysis

and then by which we have

2120588119879(119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 119891119879(119909 (119905) 119910 (119905) 119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) (119909119879(119905) 119871119879119871119909 (119905) + 119910

119879(119905) 119871119879119871119910 (119905))

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905) 119876120588 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905120588119879(119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

119909119879(119905) 119878119909 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905119909119879(119905) 119878119909 (119905)

(16)

Moreover by Lemma 2 one can get

minus int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minusint119905

119905minus120575(119905)

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(17)

Letting 1198711= 119871119879119871 and substituting (14)ndash(17) into (13) yield

119871119881 (119909 (119905) 119910 (119905) 119905)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) 119909119879(119905) 1198711119909 (119905)

+ 2120582max (119875) 119910119879(119905) 1198711119910 (119905)

+ (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575)) 120588119879(119905) 119876120588 (119905)

+ 119890120573120575120588119879(119905) 1198761120588 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(18)

Furthermore from (14) it follows that

119860 = 2120578119879(119873119879

1 119873119879

2 119873119879

3 119873119879

4 119873119879

5)119879

119872 = 0

119861 = 2119909119879(119905) 119880 [120588 (119905) minus 119909 (119905) + 119866 (119910 (119905))] = 0

(19)

where 120578 = (120588119879 120588119879(119905 minus 120575(119905)) 119909119879(119905) 119910119879(119905) (int119905

119905minus120575(119905)119891(119909 119910 119904)

d119904)119879)119879 and 119873119894(1 le 119894 le 5) 119880 are matrices with compatible

dimensionsIt can be shown that

int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573119905(119860 + 119861) + 119890

1205731199051198721198791198751(119872(119905) + int

119905

119905minus120575(119905)

119892 (119909 119910 119904) d119904)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 119909119879(119905) 1198711119909 (119905) + 119890

120573120575120588119879(119905) 119876120588 (119905)

+ 119910119879(119905) 1198711119910 (119905) + 2120582max (119875) 119909

119879(119905) 1198711119909 (119905)

+ 2120582max119910119879(119905) 1198711119910 (119905)

minus (1 minus 120575) 120588119879(119905) 119876120588 (119905) + 119890

120573120575119909119879(119905) 119878119909 (119905)

minus (1 minus 120575) 119909119879(119905) 119878119909 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

le 119890120573119905120578119879Ω120578 + 120576

minus1119909119879(119905) 119880119880

119879119909 (119905)

(20)

whereΩ is defined as

Ω =

((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

5

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

5

lowast lowast Ξ33

0 minus119873119879

3

lowast lowast lowast Ξ44

minus119873119879

4

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

5

))))))))

)

(21)

Abstract and Applied Analysis 5

By Schur complement we know that 120578119879Ω120578 +

120576minus1119909119879(119905)119880119880

119879119909(119905) lt 0 On the other hand it follows

that

119881 (120588 (119905) 119905) = 119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ int119905

0

2119890120573119904119909119904(119905) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(22)

Note that 120585 is bounded and119881119866 are continuous then119881(120588(0))must be nonnegative bounded Moreover L119881(119909 119910 119905) le 0

can be obtained directly

119881 (120588 (119905) 119905) le 119881 (120588 (0) 0)

+ int119905

0

2119890120573119905119909119879(119904) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(23)

By applying Lemma 2 to (23) one sees that

lim119905rarrinfin

sup119881 (120588 (119905) 119905) lt infin (24)

hence there exists a positive random variable 120577 satisfying

sup0le119905ltinfin

1198901205731199051003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))

10038161003816100381610038162

le 120577 (25)

Since for any 1205763isin (0 1)

1003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))10038161003816100381610038162

ge (1 minus 120576minus1

3) |119909 (119905)|

2minus (1205763minus 1)

1003816100381610038161003816119866 (119910 (119905))10038161003816100381610038162

(26)

we must have

sup0le119905le119879

119890120573119905|119909 (119905)|

2

le 120577 +1198962

1205763

sup0le119905le119879

119890120573119905 1003816100381610038161003816119910 (119905)

10038161003816100381610038162

le 120577 + 11989621198901205731205911003817100381710038171003817120585

10038171003817100381710038172

+1198962

1205763

119890120573120591 sup0le119905le119879

119890120573119905|119909 (119905)|

2

(27)

From the above inequality (26) it yields the desired result

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus

120573

2 (28)

That completes the proof

4 Example

In this section a numerical example will be given to illustratethat the proposed method is effective

Example 1 Consider the following system

119889 [1199091(119905) minus 01119909

2(119905 minus 120575 (119905))] = minus 119909

1(119905) 1199092(119905 minus 120575 (119905)) d119905

+ 1199091(119905) sin2 (119905 minus 120575 (119905)) d120596 (119905)

119889 [1199092(119905) minus 01119909

1(119905 minus 120575 (119905))] = minus 119909

2(119905) 1199091(119905 minus 120575 (119905)) d119905

+ 1199092(119905) cos2 (119905 minus 120575 (119905)) d120596 (119905)

(29)

where the delay function is defined as 120575(119905) = (14) sin(119905) 119905 gt0 It is obvious that (29) satisfies the assumptions H

1and H

2

and here 119871 = 119868 119896 = 01 Moreover since 120575 = (14) cos(119905)then 120575 = 14

According to Theorem 4 and employing MATLAB LMIToolbox it is relatively easy to deduce that the neutralstochastic differential equation (29) is almost sure exponen-tially stable

Remark 5 Comparing with some existing sufficient criteriafor neural stochastic differential equations (eg [6 11 12])the obtained result is given in terms of linear matrix inequal-ity (LMI) which can be easily checked by MATLAB LMIToolbox

5 Conclusion

The exponential stability is investigated for a class of neutralstochastic differential equations with time-varying delaysIn order to overcome the difficulties we introduce suitableLyapunov functionals and employ linear matrix inequalities(LMIs) technique and then a delay-dependent criteria aregiven to check the almost sure exponential stability of theconcerned equations

Conflict of Interests

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

References

[1] J K Hale and K R Meyer A Class of Functional Equations ofNeutral Type Memoirs of the American Mathematical Societyno 76 American Mathematical Society Providence RI USA1967

[2] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[3] V B Kolmanovskii and V R Nosov Stability of Functional-Differential Equations vol 180 of Mathematics in Science andEngineering Academic Press New York NY USA 1986

[4] V B Kolmanovskii and A Myshkis Applied Theory of Func-tional Differential Equations Kluwer Academic PublishersNorwell Mass USA 1992

[5] X Mao ldquoExponential stability in mean square of neutralstochastic differential-functional equationsrdquo Systems amp ControlLetters vol 26 no 4 pp 245ndash251 1995

[6] Q Luo X Mao and Y Shen ldquoNew criteria on exponentialstability of neutral stochastic differential delay equationsrdquoSystems amp Control Letters vol 55 no 10 pp 826ndash834 2006

[7] V B Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999

[8] M SMahmoud ldquoRobust119867infincontrol of linear neutral systemsrdquo

Automatica vol 36 no 5 pp 757ndash764 2000[9] X Mao ldquoRazumikhin-type theorems on exponential stability

of neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

6 Abstract and Applied Analysis

[10] XMao ldquoAsymptotic properties of neutral stochastic differentialdelay equationsrdquo Stochastics and Stochastics Reports vol 68 no3-4 pp 273ndash295 2000

[11] M Milosevic ldquoHighly nonlinear neutral stochastic differentialequations with time-dependent delay and the Euler-MaruyamamethodrdquoMathematical and Computer Modelling vol 54 no 9-10 pp 2235ndash2251 2011

[12] M Milosevic ldquoAlmost sure exponential stability of solutions tohighly nonlinear neutral stochastic differential equations withtime-dependent delay and the Euler-Maruyama approxima-tionrdquo Mathematical and Computer Modelling vol 57 no 3-4pp 887ndash899 2013

[13] S He ldquoResilient 1198712-119871infinfiltering of uncertain Markovian jump-

ing systemswithin the finite-time intervalrdquoAbstract andAppliedAnalysis vol 2013 Article ID 791296 7 pages 2013

[14] S He and F Liu ldquoAdaptive observer-based fault estimation forstochastic Markovian jumping systemsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 176419 11 pages 2012

[15] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198712minus119871infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[16] H Shen Y Chu S Xu and Z Zhang ldquoDelay-dependent119867infin

control for jumping delayed systems with two markovprocessesrdquo International Journal of Control Automation andSystems vol 9 no 3 pp 437ndash441 2011

[17] H Shen J Park L Zhang and Z Wu ldquoRobust extendeddissipative control for sampled-data Markov jump systemsrdquoInternational Journal of Control 2014

[18] J ChengH Zhu S Zhong Y Zhang andY Li ldquoFinite-time119867infin

control for a class of discrete-time Markov jump systems withpartly unknown time-varying transition probabilities subject toaverage dwell time switchingrdquo Journal of Systems Science 2013

[19] J Cheng H Zhu S Zhong Y Zeng and L Hou ldquoFinite-time 119867

infinfiltering for a class of discrete-time Markovian jump

systems with partly unknown transition probabilitiesrdquo Interna-tional Journal of Adaptive Control and Signal Processing 2013

[20] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 pp 768ndash774 2013

[21] S He and F Liu ldquoFinite-time 119867infin

control of nonlinear jumpsystems with timedelays via dynamic observer-based statefeedbackrdquo IEEE Transactions on Fuzzy Systems vol 20 no 4pp 605ndash614 2012

[22] S-P He and F Liu ldquoRobust finite-time119867infincontrol of stochastic

jump systemsrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1336ndash1341 2010

[23] R Sh Liptser and A N Shiryayev Theory of Martingales vol49 of Mathematics and Its Applications (Soviet Series) KluwerAcademic Publishers Dordrecht The Netherlands 1989

[24] K Gu ldquoAn integral inequality in the stability problem oftime-delay systemsrdquo in Proceedings of the 39th IEEE Confernceon Decision and Control pp 2805ndash2810 Sydney AustraliaDecember 2000

[25] Q Luo and Y Zhang ldquoAlmost sure exponential stability ofstochastic reaction diffusion systemsrdquo Nonlinear Analysis The-ory Methods ampApplications vol 71 no 12 pp e487ndashe493 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

and then by which we have

2120588119879(119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 119891119879(119909 (119905) 119910 (119905) 119905) 119875119891 (119909 (119905) 119910 (119905) 119905)

le 120588119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) (119909119879(119905) 119871119879119871119909 (119905) + 119910

119879(119905) 119871119879119871119910 (119905))

minus 119890120573(119905minus120575(119905))+120575

120588119879(119905) 119876120588 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905120588119879(119905) 119876120588 (119905)

minus 119890120573(119905minus120575(119905))+120575

119909119879(119905) 119878119909 (119905) (1 minus 120575 (119905))

le minus (1 minus 120575) 119890120573119905119909119879(119905) 119878119909 (119905)

(16)

Moreover by Lemma 2 one can get

minus int119905

119905minus120575

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minusint119905

119905minus120575(119905)

119891119879(119909 (119904) 119910 (119904) 119904) 119877119891 (119909 (119904) 119910 (119904) 119904) d119904

le minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(17)

Letting 1198711= 119871119879119871 and substituting (14)ndash(17) into (13) yield

119871119881 (119909 (119905) 119910 (119905) 119905)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 2120582max (119875) 119909119879(119905) 1198711119909 (119905)

+ 2120582max (119875) 119910119879(119905) 1198711119910 (119905)

+ (minus (1 minus 120575) 119890120573120575

or minus (1 minus 120575)) 120588119879(119905) 119876120588 (119905)

+ 119890120573120575120588119879(119905) 1198761120588 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

(18)

Furthermore from (14) it follows that

119860 = 2120578119879(119873119879

1 119873119879

2 119873119879

3 119873119879

4 119873119879

5)119879

119872 = 0

119861 = 2119909119879(119905) 119880 [120588 (119905) minus 119909 (119905) + 119866 (119910 (119905))] = 0

(19)

where 120578 = (120588119879 120588119879(119905 minus 120575(119905)) 119909119879(119905) 119910119879(119905) (int119905

119905minus120575(119905)119891(119909 119910 119904)

d119904)119879)119879 and 119873119894(1 le 119894 le 5) 119880 are matrices with compatible

dimensionsIt can be shown that

int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ 119890120573119905(119860 + 119861) + 119890

1205731199051198721198791198751(119872(119905) + int

119905

119905minus120575(119905)

119892 (119909 119910 119904) d119904)

le 119890120573119905120573120588119879(119905) 119875120588 (119905) + 120588

119879(119905) 119875119879120588 (119905)

+ 119909119879(119905) 1198711119909 (119905) + 119890

120573120575120588119879(119905) 119876120588 (119905)

+ 119910119879(119905) 1198711119910 (119905) + 2120582max (119875) 119909

119879(119905) 1198711119909 (119905)

+ 2120582max119910119879(119905) 1198711119910 (119905)

minus (1 minus 120575) 120588119879(119905) 119876120588 (119905) + 119890

120573120575119909119879(119905) 119878119909 (119905)

minus (1 minus 120575) 119909119879(119905) 119878119909 (119905) + 2120582max

1

120573(119890120573120575

minus 1)

times (119909119879(119905) 1198711119909 (119905) + 119910

119879(119905) 1198711119910 (119905))

minus1

120575(int119905

119905minus120575(119905)

119891(119909(119904) 119910(119904) 119904)d119904)119879

times 119877(int119905

119905minus120575(119905)

119891 (119909 (119904) 119910 (119904) 119904) d119904)

le 119890120573119905120578119879Ω120578 + 120576

minus1119909119879(119905) 119880119880

119879119909 (119905)

(20)

whereΩ is defined as

Ω =

((((((((

(

Ξ11

119873119879

2minus 119873119879

1119873119879

3+ 119880119879

119873119879

4minus119873119879

1+ 119873119879

5

lowast Ξ22

minus119873119879

3minus119873119879

4minus119873119879

2minus 119873119879

5

lowast lowast Ξ33

0 minus119873119879

3

lowast lowast lowast Ξ44

minus119873119879

4

lowast lowast lowast lowast minus1

120575119877 minus 2119873

119879

5

))))))))

)

(21)

Abstract and Applied Analysis 5

By Schur complement we know that 120578119879Ω120578 +

120576minus1119909119879(119905)119880119880

119879119909(119905) lt 0 On the other hand it follows

that

119881 (120588 (119905) 119905) = 119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ int119905

0

2119890120573119904119909119904(119905) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(22)

Note that 120585 is bounded and119881119866 are continuous then119881(120588(0))must be nonnegative bounded Moreover L119881(119909 119910 119905) le 0

can be obtained directly

119881 (120588 (119905) 119905) le 119881 (120588 (0) 0)

+ int119905

0

2119890120573119905119909119879(119904) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(23)

By applying Lemma 2 to (23) one sees that

lim119905rarrinfin

sup119881 (120588 (119905) 119905) lt infin (24)

hence there exists a positive random variable 120577 satisfying

sup0le119905ltinfin

1198901205731199051003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))

10038161003816100381610038162

le 120577 (25)

Since for any 1205763isin (0 1)

1003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))10038161003816100381610038162

ge (1 minus 120576minus1

3) |119909 (119905)|

2minus (1205763minus 1)

1003816100381610038161003816119866 (119910 (119905))10038161003816100381610038162

(26)

we must have

sup0le119905le119879

119890120573119905|119909 (119905)|

2

le 120577 +1198962

1205763

sup0le119905le119879

119890120573119905 1003816100381610038161003816119910 (119905)

10038161003816100381610038162

le 120577 + 11989621198901205731205911003817100381710038171003817120585

10038171003817100381710038172

+1198962

1205763

119890120573120591 sup0le119905le119879

119890120573119905|119909 (119905)|

2

(27)

From the above inequality (26) it yields the desired result

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus

120573

2 (28)

That completes the proof

4 Example

In this section a numerical example will be given to illustratethat the proposed method is effective

Example 1 Consider the following system

119889 [1199091(119905) minus 01119909

2(119905 minus 120575 (119905))] = minus 119909

1(119905) 1199092(119905 minus 120575 (119905)) d119905

+ 1199091(119905) sin2 (119905 minus 120575 (119905)) d120596 (119905)

119889 [1199092(119905) minus 01119909

1(119905 minus 120575 (119905))] = minus 119909

2(119905) 1199091(119905 minus 120575 (119905)) d119905

+ 1199092(119905) cos2 (119905 minus 120575 (119905)) d120596 (119905)

(29)

where the delay function is defined as 120575(119905) = (14) sin(119905) 119905 gt0 It is obvious that (29) satisfies the assumptions H

1and H

2

and here 119871 = 119868 119896 = 01 Moreover since 120575 = (14) cos(119905)then 120575 = 14

According to Theorem 4 and employing MATLAB LMIToolbox it is relatively easy to deduce that the neutralstochastic differential equation (29) is almost sure exponen-tially stable

Remark 5 Comparing with some existing sufficient criteriafor neural stochastic differential equations (eg [6 11 12])the obtained result is given in terms of linear matrix inequal-ity (LMI) which can be easily checked by MATLAB LMIToolbox

5 Conclusion

The exponential stability is investigated for a class of neutralstochastic differential equations with time-varying delaysIn order to overcome the difficulties we introduce suitableLyapunov functionals and employ linear matrix inequalities(LMIs) technique and then a delay-dependent criteria aregiven to check the almost sure exponential stability of theconcerned equations

Conflict of Interests

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

References

[1] J K Hale and K R Meyer A Class of Functional Equations ofNeutral Type Memoirs of the American Mathematical Societyno 76 American Mathematical Society Providence RI USA1967

[2] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[3] V B Kolmanovskii and V R Nosov Stability of Functional-Differential Equations vol 180 of Mathematics in Science andEngineering Academic Press New York NY USA 1986

[4] V B Kolmanovskii and A Myshkis Applied Theory of Func-tional Differential Equations Kluwer Academic PublishersNorwell Mass USA 1992

[5] X Mao ldquoExponential stability in mean square of neutralstochastic differential-functional equationsrdquo Systems amp ControlLetters vol 26 no 4 pp 245ndash251 1995

[6] Q Luo X Mao and Y Shen ldquoNew criteria on exponentialstability of neutral stochastic differential delay equationsrdquoSystems amp Control Letters vol 55 no 10 pp 826ndash834 2006

[7] V B Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999

[8] M SMahmoud ldquoRobust119867infincontrol of linear neutral systemsrdquo

Automatica vol 36 no 5 pp 757ndash764 2000[9] X Mao ldquoRazumikhin-type theorems on exponential stability

of neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

6 Abstract and Applied Analysis

[10] XMao ldquoAsymptotic properties of neutral stochastic differentialdelay equationsrdquo Stochastics and Stochastics Reports vol 68 no3-4 pp 273ndash295 2000

[11] M Milosevic ldquoHighly nonlinear neutral stochastic differentialequations with time-dependent delay and the Euler-MaruyamamethodrdquoMathematical and Computer Modelling vol 54 no 9-10 pp 2235ndash2251 2011

[12] M Milosevic ldquoAlmost sure exponential stability of solutions tohighly nonlinear neutral stochastic differential equations withtime-dependent delay and the Euler-Maruyama approxima-tionrdquo Mathematical and Computer Modelling vol 57 no 3-4pp 887ndash899 2013

[13] S He ldquoResilient 1198712-119871infinfiltering of uncertain Markovian jump-

ing systemswithin the finite-time intervalrdquoAbstract andAppliedAnalysis vol 2013 Article ID 791296 7 pages 2013

[14] S He and F Liu ldquoAdaptive observer-based fault estimation forstochastic Markovian jumping systemsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 176419 11 pages 2012

[15] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198712minus119871infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[16] H Shen Y Chu S Xu and Z Zhang ldquoDelay-dependent119867infin

control for jumping delayed systems with two markovprocessesrdquo International Journal of Control Automation andSystems vol 9 no 3 pp 437ndash441 2011

[17] H Shen J Park L Zhang and Z Wu ldquoRobust extendeddissipative control for sampled-data Markov jump systemsrdquoInternational Journal of Control 2014

[18] J ChengH Zhu S Zhong Y Zhang andY Li ldquoFinite-time119867infin

control for a class of discrete-time Markov jump systems withpartly unknown time-varying transition probabilities subject toaverage dwell time switchingrdquo Journal of Systems Science 2013

[19] J Cheng H Zhu S Zhong Y Zeng and L Hou ldquoFinite-time 119867

infinfiltering for a class of discrete-time Markovian jump

systems with partly unknown transition probabilitiesrdquo Interna-tional Journal of Adaptive Control and Signal Processing 2013

[20] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 pp 768ndash774 2013

[21] S He and F Liu ldquoFinite-time 119867infin

control of nonlinear jumpsystems with timedelays via dynamic observer-based statefeedbackrdquo IEEE Transactions on Fuzzy Systems vol 20 no 4pp 605ndash614 2012

[22] S-P He and F Liu ldquoRobust finite-time119867infincontrol of stochastic

jump systemsrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1336ndash1341 2010

[23] R Sh Liptser and A N Shiryayev Theory of Martingales vol49 of Mathematics and Its Applications (Soviet Series) KluwerAcademic Publishers Dordrecht The Netherlands 1989

[24] K Gu ldquoAn integral inequality in the stability problem oftime-delay systemsrdquo in Proceedings of the 39th IEEE Confernceon Decision and Control pp 2805ndash2810 Sydney AustraliaDecember 2000

[25] Q Luo and Y Zhang ldquoAlmost sure exponential stability ofstochastic reaction diffusion systemsrdquo Nonlinear Analysis The-ory Methods ampApplications vol 71 no 12 pp e487ndashe493 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

By Schur complement we know that 120578119879Ω120578 +

120576minus1119909119879(119905)119880119880

119879119909(119905) lt 0 On the other hand it follows

that

119881 (120588 (119905) 119905) = 119881 (120588 (0) 0) + int119905

0

L119881 (119909 (119904) 119910 (119904) 119904) d119904

+ int119905

0

2119890120573119904119909119904(119905) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(22)

Note that 120585 is bounded and119881119866 are continuous then119881(120588(0))must be nonnegative bounded Moreover L119881(119909 119910 119905) le 0

can be obtained directly

119881 (120588 (119905) 119905) le 119881 (120588 (0) 0)

+ int119905

0

2119890120573119905119909119879(119904) 119892 (119909 (119904) 119910 (119904) 119904) d119908 (119904)

(23)

By applying Lemma 2 to (23) one sees that

lim119905rarrinfin

sup119881 (120588 (119905) 119905) lt infin (24)

hence there exists a positive random variable 120577 satisfying

sup0le119905ltinfin

1198901205731199051003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))

10038161003816100381610038162

le 120577 (25)

Since for any 1205763isin (0 1)

1003816100381610038161003816119909 (119905) minus 119866 (119910 (119905))10038161003816100381610038162

ge (1 minus 120576minus1

3) |119909 (119905)|

2minus (1205763minus 1)

1003816100381610038161003816119866 (119910 (119905))10038161003816100381610038162

(26)

we must have

sup0le119905le119879

119890120573119905|119909 (119905)|

2

le 120577 +1198962

1205763

sup0le119905le119879

119890120573119905 1003816100381610038161003816119910 (119905)

10038161003816100381610038162

le 120577 + 11989621198901205731205911003817100381710038171003817120585

10038171003817100381710038172

+1198962

1205763

119890120573120591 sup0le119905le119879

119890120573119905|119909 (119905)|

2

(27)

From the above inequality (26) it yields the desired result

lim sup119905rarrinfin

1

119905log |119909 (119905)| le minus

120573

2 (28)

That completes the proof

4 Example

In this section a numerical example will be given to illustratethat the proposed method is effective

Example 1 Consider the following system

119889 [1199091(119905) minus 01119909

2(119905 minus 120575 (119905))] = minus 119909

1(119905) 1199092(119905 minus 120575 (119905)) d119905

+ 1199091(119905) sin2 (119905 minus 120575 (119905)) d120596 (119905)

119889 [1199092(119905) minus 01119909

1(119905 minus 120575 (119905))] = minus 119909

2(119905) 1199091(119905 minus 120575 (119905)) d119905

+ 1199092(119905) cos2 (119905 minus 120575 (119905)) d120596 (119905)

(29)

where the delay function is defined as 120575(119905) = (14) sin(119905) 119905 gt0 It is obvious that (29) satisfies the assumptions H

1and H

2

and here 119871 = 119868 119896 = 01 Moreover since 120575 = (14) cos(119905)then 120575 = 14

According to Theorem 4 and employing MATLAB LMIToolbox it is relatively easy to deduce that the neutralstochastic differential equation (29) is almost sure exponen-tially stable

Remark 5 Comparing with some existing sufficient criteriafor neural stochastic differential equations (eg [6 11 12])the obtained result is given in terms of linear matrix inequal-ity (LMI) which can be easily checked by MATLAB LMIToolbox

5 Conclusion

The exponential stability is investigated for a class of neutralstochastic differential equations with time-varying delaysIn order to overcome the difficulties we introduce suitableLyapunov functionals and employ linear matrix inequalities(LMIs) technique and then a delay-dependent criteria aregiven to check the almost sure exponential stability of theconcerned equations

Conflict of Interests

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

References

[1] J K Hale and K R Meyer A Class of Functional Equations ofNeutral Type Memoirs of the American Mathematical Societyno 76 American Mathematical Society Providence RI USA1967

[2] J K Hale and S M V Lunel Introduction to FunctionalDifferential Equations Springer Berlin Germany 1991

[3] V B Kolmanovskii and V R Nosov Stability of Functional-Differential Equations vol 180 of Mathematics in Science andEngineering Academic Press New York NY USA 1986

[4] V B Kolmanovskii and A Myshkis Applied Theory of Func-tional Differential Equations Kluwer Academic PublishersNorwell Mass USA 1992

[5] X Mao ldquoExponential stability in mean square of neutralstochastic differential-functional equationsrdquo Systems amp ControlLetters vol 26 no 4 pp 245ndash251 1995

[6] Q Luo X Mao and Y Shen ldquoNew criteria on exponentialstability of neutral stochastic differential delay equationsrdquoSystems amp Control Letters vol 55 no 10 pp 826ndash834 2006

[7] V B Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999

[8] M SMahmoud ldquoRobust119867infincontrol of linear neutral systemsrdquo

Automatica vol 36 no 5 pp 757ndash764 2000[9] X Mao ldquoRazumikhin-type theorems on exponential stability

of neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

6 Abstract and Applied Analysis

[10] XMao ldquoAsymptotic properties of neutral stochastic differentialdelay equationsrdquo Stochastics and Stochastics Reports vol 68 no3-4 pp 273ndash295 2000

[11] M Milosevic ldquoHighly nonlinear neutral stochastic differentialequations with time-dependent delay and the Euler-MaruyamamethodrdquoMathematical and Computer Modelling vol 54 no 9-10 pp 2235ndash2251 2011

[12] M Milosevic ldquoAlmost sure exponential stability of solutions tohighly nonlinear neutral stochastic differential equations withtime-dependent delay and the Euler-Maruyama approxima-tionrdquo Mathematical and Computer Modelling vol 57 no 3-4pp 887ndash899 2013

[13] S He ldquoResilient 1198712-119871infinfiltering of uncertain Markovian jump-

ing systemswithin the finite-time intervalrdquoAbstract andAppliedAnalysis vol 2013 Article ID 791296 7 pages 2013

[14] S He and F Liu ldquoAdaptive observer-based fault estimation forstochastic Markovian jumping systemsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 176419 11 pages 2012

[15] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198712minus119871infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[16] H Shen Y Chu S Xu and Z Zhang ldquoDelay-dependent119867infin

control for jumping delayed systems with two markovprocessesrdquo International Journal of Control Automation andSystems vol 9 no 3 pp 437ndash441 2011

[17] H Shen J Park L Zhang and Z Wu ldquoRobust extendeddissipative control for sampled-data Markov jump systemsrdquoInternational Journal of Control 2014

[18] J ChengH Zhu S Zhong Y Zhang andY Li ldquoFinite-time119867infin

control for a class of discrete-time Markov jump systems withpartly unknown time-varying transition probabilities subject toaverage dwell time switchingrdquo Journal of Systems Science 2013

[19] J Cheng H Zhu S Zhong Y Zeng and L Hou ldquoFinite-time 119867

infinfiltering for a class of discrete-time Markovian jump

systems with partly unknown transition probabilitiesrdquo Interna-tional Journal of Adaptive Control and Signal Processing 2013

[20] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 pp 768ndash774 2013

[21] S He and F Liu ldquoFinite-time 119867infin

control of nonlinear jumpsystems with timedelays via dynamic observer-based statefeedbackrdquo IEEE Transactions on Fuzzy Systems vol 20 no 4pp 605ndash614 2012

[22] S-P He and F Liu ldquoRobust finite-time119867infincontrol of stochastic

jump systemsrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1336ndash1341 2010

[23] R Sh Liptser and A N Shiryayev Theory of Martingales vol49 of Mathematics and Its Applications (Soviet Series) KluwerAcademic Publishers Dordrecht The Netherlands 1989

[24] K Gu ldquoAn integral inequality in the stability problem oftime-delay systemsrdquo in Proceedings of the 39th IEEE Confernceon Decision and Control pp 2805ndash2810 Sydney AustraliaDecember 2000

[25] Q Luo and Y Zhang ldquoAlmost sure exponential stability ofstochastic reaction diffusion systemsrdquo Nonlinear Analysis The-ory Methods ampApplications vol 71 no 12 pp e487ndashe493 2009

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

6 Abstract and Applied Analysis

[10] XMao ldquoAsymptotic properties of neutral stochastic differentialdelay equationsrdquo Stochastics and Stochastics Reports vol 68 no3-4 pp 273ndash295 2000

[11] M Milosevic ldquoHighly nonlinear neutral stochastic differentialequations with time-dependent delay and the Euler-MaruyamamethodrdquoMathematical and Computer Modelling vol 54 no 9-10 pp 2235ndash2251 2011

[12] M Milosevic ldquoAlmost sure exponential stability of solutions tohighly nonlinear neutral stochastic differential equations withtime-dependent delay and the Euler-Maruyama approxima-tionrdquo Mathematical and Computer Modelling vol 57 no 3-4pp 887ndash899 2013

[13] S He ldquoResilient 1198712-119871infinfiltering of uncertain Markovian jump-

ing systemswithin the finite-time intervalrdquoAbstract andAppliedAnalysis vol 2013 Article ID 791296 7 pages 2013

[14] S He and F Liu ldquoAdaptive observer-based fault estimation forstochastic Markovian jumping systemsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 176419 11 pages 2012

[15] Z Wu P Shi H Su and J Chu ldquoAsynchronous 1198712minus119871infin

filtering for discrete-time stochastic Markov jump systems withrandomly occurred sensor nonlinearitiesrdquo Automatica vol 50no 1 pp 180ndash186 2014

[16] H Shen Y Chu S Xu and Z Zhang ldquoDelay-dependent119867infin

control for jumping delayed systems with two markovprocessesrdquo International Journal of Control Automation andSystems vol 9 no 3 pp 437ndash441 2011

[17] H Shen J Park L Zhang and Z Wu ldquoRobust extendeddissipative control for sampled-data Markov jump systemsrdquoInternational Journal of Control 2014

[18] J ChengH Zhu S Zhong Y Zhang andY Li ldquoFinite-time119867infin

control for a class of discrete-time Markov jump systems withpartly unknown time-varying transition probabilities subject toaverage dwell time switchingrdquo Journal of Systems Science 2013

[19] J Cheng H Zhu S Zhong Y Zeng and L Hou ldquoFinite-time 119867

infinfiltering for a class of discrete-time Markovian jump

systems with partly unknown transition probabilitiesrdquo Interna-tional Journal of Adaptive Control and Signal Processing 2013

[20] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-

dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 pp 768ndash774 2013

[21] S He and F Liu ldquoFinite-time 119867infin

control of nonlinear jumpsystems with timedelays via dynamic observer-based statefeedbackrdquo IEEE Transactions on Fuzzy Systems vol 20 no 4pp 605ndash614 2012

[22] S-P He and F Liu ldquoRobust finite-time119867infincontrol of stochastic

jump systemsrdquo International Journal of Control Automation andSystems vol 8 no 6 pp 1336ndash1341 2010

[23] R Sh Liptser and A N Shiryayev Theory of Martingales vol49 of Mathematics and Its Applications (Soviet Series) KluwerAcademic Publishers Dordrecht The Netherlands 1989

[24] K Gu ldquoAn integral inequality in the stability problem oftime-delay systemsrdquo in Proceedings of the 39th IEEE Confernceon Decision and Control pp 2805ndash2810 Sydney AustraliaDecember 2000

[25] Q Luo and Y Zhang ldquoAlmost sure exponential stability ofstochastic reaction diffusion systemsrdquo Nonlinear Analysis The-ory Methods ampApplications vol 71 no 12 pp e487ndashe493 2009

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