Research ArticleA New Stability Criterion for Neutral Stochastic DelayDifferential Equations with Markovian Switching

Wei Hu 12

1School of Mathematics and Physics Jiangsu University of Technology Changzhou 213001 Jiangsu China2School of Mathematical Sciences and Institute of Finance and Statistics Nanjing Normal University Nanjing 210023 Jiangsu China

Correspondence should be addressed to Wei Hu huwjsuteducn

Received 22 July 2018 Accepted 25 September 2018 Published 14 October 2018

Academic Editor Ju H Park

Copyright copy 2018 Wei Hu This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this short paper a new stability theorem for neutral stochastic delay differential equations with Markovian switching isinvestigated by applying stochastic analysis technique and Razumikhin stability approach A novel criterion of the 119901th momentexponential stability is derived for the related systems The feature of the criterion shows that the estimated upper bound for thediffusion operator of Lyapunov function is allowed to be indefinite even if to be unbounded which can loosen the constraints ofthe existing results Last an example is provided to illustrate the usefulness and significance of the theoretical results

1 Introduction

As is well-known the stability is very hot topic in deter-ministic or stochastic dynamic systems (for instance see [1ndash37] and references therein) Recently a class of stochastichybrid systems (also known as stochastic systems withMarkovian switching) has been used tomodelmany practicalsystems where they may experience abrupt changes in theirstructure and parameters In [38] the author explained thatthe Markovian switching systems had been emerging as aconvenient mathematical framework for the formulation ofvarious design problems in different fields such as targettracking fault tolerant control andmanufacturing processeswhich can be seen as the motivation of wide practical useof the theoretic results for hybrid systems Owing to theirwidely real applications stochastic systems with Markovianswitching have received a great deal of attention and manyinteresting results have been reported in the literature Forexample [39] considered the stability of stochastic differ-ential equations with infinite Markovian switchings and[40] studied the stability for stochastic differential equationswith semi-Markovian switchings The results of stochas-tic hybrid systems are applied in feedback controls andneural networks please refer to [41ndash45] and the referencetherein

As an important class of hybrid stochastic system neu-tral stochastic delay differential equations with Markovianswitching (NSDDEwMS) have been applied in practice suchas traffic control neutral networks and chemical processDue to itswide applicationsmore andmore researchers focuson this system An important question for such a modelis the stability analysis Lyapunov-Razumikhin functionsmethod and Lyapunov-Krasovskii functionals method aretwo effective methods that have been exploited for the stabil-ity analysis For exampleMao in [46 47] appliedRazumikhinapproach to derive the exponential stability criteria forneutral stochastic delay differential systems with Markovianswitching In [48ndash53] the authors considered the exponentialstability by using Lyapunov-Krasovskii functionals methodReference [54] applied some special techniques to studythe exponential stability Besides Zhu in [55 56] obtainedseveral novel exponential stability criteria for some morecomplex systems with impulse control and Levy noise Inaddition [57] considered the stability in distribution of neu-tral stochastic differential delay equations with Markovianswitching and [58 59] studied the almost sure stability for thesame system

Although the above exponential stability criteria can beused to judge the stability for many NSDDEwMS neverthe-less these conditions in the above literature can be weakened

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7814974 8 pageshttpsdoiorg10115520187814974

2 Mathematical Problems in Engineering

to cover a large collection of NSDDEwMS In other wordsthere are some shortcomings in the former criteria Themain is that all the above criteria for exponential stabilityin the related literature imposed some strict conditions onthe diffusion operator of Lyapunov functions For example[46 47] required the estimated upper bound for the diffusionoperator of Lyapunov functions to be negative constant num-bers Reference [48] needed that the estimated upper bound120582(119905) must be a negative value function Simply speakingthe former results all need that the upper bound 120582(119905) forthe diffusion operator of the Lyapunov function is negativedefinite for all 119905 This restriction leads to the criteria havingstrong conservativeness in practice due to the fact that thereare a large number of time-varying systems not satisfying theabove conditions As shown by an example in Section 4 theexisting results and methods cannot be applied to analyse thestability for more general time-varying systems

Motivated by the above discussion in this short notewe focus on the exponential stability for NSDDEwMS Byusing the notions such as uniformly stable function (USF)overshoot of the USF that dated from [60 61] and thencombining the stochastic analysis techniques we obtain anovel stability criterion for NSDDEwMS The feature of thecriterion is that the coefficients of the estimated upper boundfor the diffusion operator of Lyapunov functions can beallowed sign-changing as the time-varying which can beused more widely than the existing results

The rest of the paper is organized as follows In Section 2we introduce the model of NSDDEwMS and some prelim-inaries The main result and its proof will be displayed inSection 3 An example is provided in Section 4 Finally weconclude this paper with some general remarks in Section 5

Throughout this short paper let (ΩF F119905119905ge0P) be acomplete probability space with a natural filtration F119905119905ge0satisfying the usual condition (ie it is right continuousand F0 contains all P-null sets) |119909| denotes the Euclideannorm of 119909 The symbol 119862([minus120591 0]R119899) denotes the family ofcontinuous function 120593 from [minus120591 0] to R119899 with the norm120593 = supminus120591le120579le0|120593(120579)| We use 119862119901

F119905([minus120591 0]R119899) to denote the

family of all F119905-measurable 119862([minus120591 0]R119899) valued randomvariables 120593 satisfying supminus120591le120579le0E|120593(120579)|119901 lt infin We use E[sdot] todenote the expectation operator with respect to the probabil-ity measure P Let 119861119905 = 119861(119905) = (1198611(119905) 1198612(119905) 119861119898(119905))119879 bean 119898-dimensional Brownian motion defined on a completeprobability space 119886 or 119887 = max119886 119887 N denotes the set ofpositive integers

2 Preliminaries

Let 119903(119905) 119905 ge 0 be a right-continuous Markov chain on theprobability space (ΩFP) taking values in a finite state spaceS = 1 2 119873 with generator 119876 = (119902119894119895)119873times119873 given by

P (119903 (119905 + Δ119905) = 119895 | 119903 (119905) = 119894)

= 119902119894119895Δ119905 + 119900 (Δ119905) 119894119891 119894 = 1198951 + 119902119894119894Δ119905 + 119900 (Δ119905) 119894119891 119894 = 119895

(1)

where Δ119905 gt 0 Here 119902119894119895 ge 0 is the transition rate from 119894 to 119895 if119894 = 119895 while 119902119894119894 = minussum119895 =119894 119902119894119895 = minus119902119894In this short paper we will consider the following neutral

stochastic delay differential system with Markovian switch-ing

119889 [119909 (119905) minus 119906 (119905 119909 (119905 minus 120591) 119903 (119905))]= 119891 (119905 119909 (119905) 119909 (119905 minus 120591) 119903 (119905)) 119889119905

+ 119892 (119905 119909 (119905) 119909 (119905 minus 120591) 119903 (119905)) 119889119861119905(2)

with the initial data 1199090 = 120601 = 120601(120579) minus120591 le 120579 le0 isin 119862119901F0

([minus120591 0]R119899) We assume that 119891 119892 satisfy thelocal Lipschitz condition and the linear growth conditionAdditionally we impose the following condition

1003816100381610038161003816119906 (119905 119909 119894) minus 119906 (119905 119910 119894)1003816100381610038161003816 le 120581 1003816100381610038161003816119909 minus 1199101003816100381610038161003816 120581 isin (0 1) 119894 isin S (3)

on 119906(sdot) to guarantee the existence and uniqueness of solution119909(119905 120601) for system (2) for all 119905 ge 0 We also assume that119891(119905 0 0 119894) = 0 119892(119905 0 0 119894) = 0 and 119906(119905 0 119894) = 0 whichimplies that the trivial solution of system (2) exists In order touse Lyapunovrsquos method to study the 119901th moment exponentialstability we need the following definitions

Definition 1 The trivial solution of (2) is said to be 119901thmoment exponentially stable if

lim sup119905997888rarrinfin

lnE 1003816100381610038161003816119909 (119905 120601)1003816100381610038161003816119901119905 lt 0 (4)

for all 120601 isin 119862119901F0

([minus120591 0]R119899) When 119901 = 2 2th moment expo-nential stability is usually called mean square exponentiallystable

Definition 2 The function 119881(119905 119909 119894) R+ times R119899 times S 997888rarr R+

belongs to class Ψ if it is a continuously twice differentiablewith respect to 119909 and once differentiable with respect to 119905

Next we need an operator L119881 from R+ times R119899 times R119899 times S

to R by

L119881(119905 119909 119910 119894) = 119881119905 (119905 119909 minus 119906 (119905 119910 119894) 119894) + 119881119909 (119905 119909minus 119906 (119905 119910 119894) 119894) 119891 (119905 119909 119910 119894) + 1

2119905119903119886119888119890 [119892119879 (119905 119909 119910 119894)sdot 119881119909119909 (119905 119909 minus 119906 (119905 119910 119894) 119894) 119892 (119905 119909 119910 119894)]

+ 119873sum119895=1

119902119894119895119881 (119905 119909 minus 119906 (119905 119910 119894) 119895)

(5)

Mathematical Problems in Engineering 3

where

119881119909 (119905 119909 119894)= (120597119881 (119905 119909 119894)

1205971199091 120597119881 (119905 119909 119894)1205971199092 120597119881 (119905 119909 119894)

120597119909119899 )

119881119909119909 (119905 119909 119894) = (1205972119881(119905 119909 119894)120597119909119894120597119909119895 )

119899times119899

119881119905 (119905 119909 119894) = 120597119881 (119905 119909 119894)120597119905

(6)

In order to overcome the difficult caused by the neutralitem we need the following inequality

Lemma 3 Let 119901 ge 1 Then

(1 minus 120581)119901minus1 |119909|119901 minus 120581 (1 minus 120581)119901minus1 10038161003816100381610038161199101003816100381610038161003816119901 le 1003816100381610038161003816119909 minus 119906 (119905 119910 119894)1003816100381610038161003816119901le (1 + 120581)119901minus1 (|119909|119901 + 120581 10038161003816100381610038161199101003816100381610038161003816119901)

(7)

holds for any (119905 119909 119910 119894) isin R+ timesR119899 timesR119899 timesS

The following lemma which is important in the proof ofthe main result can be found in Lemma 1 of [62]

Lemma 4 Let 119879lowast be a constant Let a function 119908 [minus119879lowastinfin) 997888rarr [0infin) admit a sequence V119894 and positiveconstants V119886 and V119887 such that V0 = 1199050 V119894+1 minus V119894 isin [V119886 V119887] forall 119894 ge 0 119908 is continuous on [V119894 V119894+1) for all 119894 ge 0 and the leftlimit lim119905997888rarrVminus119894 119908(119905) exist Assume that there exists a constant120588 isin (0 1) such that 119908(119905) le 120588 sup119904isin[119905minus119879lowast 119905]119908(119904) holds for all119905 ge 1199050 Then

119908 (119905) le 119890((ln 120588)119879lowast)(119905minus1199050) sup119904isin[1199050minus119879lowast 1199050]

119908 (119904) (8)

holds for all 119905 ge 1199050The following definitions which are dated from [60] are

important in the proof of our main result

Definition 5 A piecewise continuous function 120583 is said to beaUSF if the following linear time-varying equation is globallyuniformly asymptotically stable

119889119910 (119905) = 120583 (119905) 119910 (119905) 119889119905 forall119905 ge 0 (9)

Definition 6 Let 120583(119905) be a USF Then the set

Ω120583 = 119879 gt 0 sup119905ge0

int119905+119879119905

120583 (119904) 119889119904 lt 0 (10)

is said to be the uniform convergence set of 120583(119905) For any 119879 ge0 the overshoot 120593120583(119879) fl 120593120583 of 120583(119905) is defined as follows

120593120583 = sup119905ge0

max120579isin[0119879]

int119905+120579119905

120583 (119904) 119889119904 (11)

3 Main Results

In this section we will use stochastic analysis theory Lya-punovrsquos method and Lemmas 3 and 4 to obtain a criterionfor 119901th moment exponential stability of system (2) Ourmainresult is the following

Theorem 7 Let 119901 1198881 1198882 120581 120591 be all positive numbers and 120583(119905)is a USF If there exist a function 119881 isin Ψ and a constant 119879 isin Ω120583such that the following conditions hold for all 119905 ge 0

(1) for all 119909 isin R119899 119894 isin S

1198881 |119909|119901 le 119881 (119905 119909 119894) le 1198882 |119909|119901 (12)

(2) the following Razumikhin-type condition holds

EL119881(119905 119909 (119905) 119909 (119905 minus 120591) 119903 (119905))le 120583 (119905)E119881 (119905 119909 (119905) 119903 (119905))

if min119894isin119878

E119881(119905 + 120579 119909 (119905 + 120579) 119894)le 119902max119894isin119878

E119881(119905 119909 (119905) 119894)

forall120579 isin [minus120591 0] where 119902 gt 11988821198881 119890120593120583 (1 minus 120581)minus119901

(13)

Then system (2) is 119901th moment exponentially stable here119909(119905) ≜ 119909(119905) minus 119906(119905 119909(119905 minus 120591) 119903(119905))Proof First of all we will prove the following fact if119902minus1E119881(119905 + 120579 119909(119905 + 120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905)) holds for all120579 isin [minus120591 0] which implies EL119881(119905 119909(119905) 119909(119905 minus 120591) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) then for any 119905 gt 119879E119881 (119905 119909 (119905) 119903 (119905))

le maxE119881 (119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879)) 119890int119905119905minus119879 120583(119904)119889119904

119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881 (119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583

(14)

In order to prove inequality (14) we consider the followingtwo cases

(A) sup120579isin[minus1205910]E119881(119904 + 120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904)119903(119904)) holds for all 119904 isin [119905 minus 119879 119905](B) sup120579isin[minus1205910]E119881(119904 + 120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904)119903(119904)) does not hold for some 119904 isin [119905 minus 119879 119905]For (A) we know that EL119881(119904 119909(119904) 119909(119904 minus 120591) 119903(119904)) le120583(119904)E119881(119904 119909(119904) 119903(119904)) holds for any 119904 isin [119905 minus 119879 119905] By using

4 Mathematical Problems in Engineering

Itorsquos formula and the standard stopping times technique weobtain that

E(119890minusint119905119905minus119879 120583(119904)119889119904119881(119905 119909 (119905) 119903 (119905))) = E119881(119905 minus 119879 119909 (119905minus 119879) 119903 (119905 minus 119879))+ Eint119905119905minus119879

[minus120583 (119904) 119890minus int119904119905minus119879 120583(V)119889V119881 (119904 119909 (119904) 119903 (119904))+ 119890minus int119904119905minus119879 120583(V)119889VL119881(119904 119909 (119904) 119909 (119904 minus 120591) 119903 (119904))] 119889119904le E119881(119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879))

(15)

which implies that E119881(119905 119909(119905) 119903(119905)) le E119881(119905minus119879 119909(119905 minus119879) 119903(119905 minus119879))119890int119905119905minus119879 120583(119904)119889119904

For (B) define 119905lowast = sup119904 isin [119905 minus 119879 119905] sup120579isin[minus1205910]E119881(119904 +120579 119909(119904 + 120579) 119903(119904 + 120579)) gt 119902E119881(119904 119909(119904) 119903(119904)) Then 119905lowast lt 119905 or119905lowast = 119905 If 119905lowast lt 119905 then for all 119904 isin [119905lowast 119905] sup120579isin[minus1205910]E119881(119904 +120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904) 119903(119904)) So according to theresult of (A) we obtain that

E119881(119905 119909 (119905) 119903 (119905)) le 119890int119905119905lowast 120583(119904)119889119904E119881(119905lowast 119909 (119905lowast) 119903 (119905lowast))= 119890int119905119905lowast 120583(119904)119889119904119902minus1 sup

120579isin[minus1205910]E119881(119905lowast

+ 120579 119909 (119905lowast + 120579) 119903 (119905lowast + 120579))le 119890int119905119905lowast 120583(119904)119889119904119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904+ 120579 119909 (119904 + 120579) 119903 (119904 + 120579))le 119890120593120583119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904+ 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

(16)

If 119905lowast = 119905 from the definition of 119905lowast we haveE119881(119905 119909 (119905) 119903 (119905)) = E119881(119905lowast 119909 (119905lowast) 119903 (119905lowast))le 119902minus1 sup120579isin[minus1205910]

E119881 (119905lowast + 120579 119909 (119905lowast + 120579) 119903 (119905lowast + 120579))le 119890120593120583119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579)) (17)

Combining the above three inequalities we can see thatinequality (14) holds

Assume that 119902minus1E119881(119905+120579 119909(119905+120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905))holds for all 120579 isin [minus120591 0] then 119902minus1min119894isin119878E119881(119905 + 120579 119909(119905 +120579) 119894) le 119902minus1E119881(119905 + 120579 119909(119905 + 120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905)) lemax119894isin119878E119881(119905 119909(119905) 119894) holds for all 120579 isin [minus120591 0] According tocondition (2) we know that EL119881(119905 119909(119905) 119909(119905 minus 120591) 119903(119905)) le

120583(119905)E119881(119905 119909(119905) 119903(119905))Thus from inequality (14) condition (1)and Lemma 3 we conclude that for any 119905 gt 119879E119881(119905 119909 (119905) 119903 (119905))

le maxE119881(119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879)) 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max1198882E |119909 (119905 minus 119879)|119901 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max1198882 (1 + 120581)119901minus1

sdot (E |119909 (119905 minus 119879)|119901 + 120581E |119909 (119905 minus 119879 minus 120591)|119901) 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579119909 (119904 + 120579) 119903 (119904 + 120579))

(18)

By condition (1) we can see that

E |119909 (119905)|119901 le 11988821198881max11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

fl 120578 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 (19)

Using Lemma 3 again we obtain that

E |119909 (119905)|119901le [E |119909 (119905)|119901 + 120581 (1 minus 120581)119901minus1 E |119909 (119905 minus 120591)|119901] (1 minus 120581)1minus119901= (1 minus 120581)1minus119901 E |119909 (119905)|119901 + 120581E |119909 (119905 minus 120591)|119901le [(1 minus 120581)1minus119901 120578 + 120581] sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 (20)

In order to use Lemma 4 we need to justify that (1minus120581)1minus119901120578+120581 lt 1 In fact on one hand due to 119902 gt (11988821198881)119890120593120583(1 minus 120581)minus119901there exists a constant 120588 isin (0 1) such that (11988821198881)119902minus1119890120593120583 (1 minus120581)1minus119901 le 120588 minus 120581 On the other hand due to 120583(119905) being a USFand 119879 isin Ω120583 there exists a constant 1205881 isin (0 1) such that

Mathematical Problems in Engineering 5

119890int119905119905minus119879 120583(119904)119889119904 lt (11988811198882)2((1 minus 120581)(1 + 120581))119901((1205881 minus 120581)(1 minus 120581)) lt 1Thus

E |119909 (119905)|119901 le [(1 minus 120581)1minus119901 120578 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 = [(1 minus 120581)1minus119901 11988821198881sdotmax11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le [max 1205881 minus 120581 120588 minus 120581 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879minus120591119905]

E |119909 (119904)|119901

(21)

where 120588lowast = 120588 or 1205881 isin (0 1) By Lemma 4 we derive that

E |119909 (119905)|119901 le 119890((ln 120588lowast)(119879+120591))(119905minus119879) sup119904isin[minus120591119879]

E |119909 (119904)|119901 (22)

which implies that

lim sup119905997888rarrinfin

lnE |119909 (119905)|119901119905 = ln 120588lowast

119879 + 120591 lt 0 (23)

In other words we have proved that system (2) is119901thmomentexponentially stable

Remark 8 Here by using stochastic analysis theory somenotions as uniformly stable function (USF) overshoot ofthe USF that dated from [60 61] and some inequalitytechniques we obtain a novel exponential stability criterionwith respect to the related systems The method we use hereis rather different from the traditional Razumikhin approachThus we provide a new method to investigate the stabilityanalysis for neutral stochastic delayed differential equationswith Markovian switchings In the future we can developthis approach to study the stability for impulsive neutralstochastic delayed differential equations

Remark 9 If system (2) has no Markovian switching and120583(119905) equiv 120583 lt 0 then 119890120593120583 = the condition of Theorem 7 119902 gt(11988821198881)119890120593120583(1 minus 120581)minus119901 becomes 119902 gt (11988821198881)(1 minus 120581)minus119901 which is thesame as the conditions in [46 47] If the model we consideredhas no Markovian switching stochastic disturbance andneutral item then we can obtain the same result as Theorem1 in [60] Thus our result is the generalization of Theorem 1in [60] and the main results in [46 47]

Remark 10 In Theorem 31 of [53] the condition (33)ensured that the coefficient of the estimated upper boundfor the diffusion operator of Lyapunov function is a negative

value and the condition of Theorem 31 in [48] requiredthat the coefficient function of the estimated upper boundfor the diffusion operator of Lyapunov function is a negativedefinite function which are very conservative in the practiceif the system is time-varying system But our conditions allowthis function to take values on R and even allow it to beunbounded Thus our result can be used more widely and isless conservative than the existing results See Example 1 formore details

4 An Example

In this section we will give an example to illustrate thevalidity and significance of our result

Example 1 Now we will consider the following neutral-type stochastic delay differential system with Markovianswitching

119889 [119909 (119905) minus 02119909 (119905 minus 05)]= 119891 (119905 119909 (119905) 119909 (t minus 05) 119903 (119905)) 119889119905

+ 119892 (119905 119909 (119905) 119909 (119905 minus 05) 119903 (119905)) 119889119861119905(24)

where119891 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

=

( 116119905 cos 1199052 minus

32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905) 119894 = 1(18119905 cos 1199052 minus 1) 119909 (119905 minus 05) 119894 = 2

(25)

and119892 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

= (tanh 119905) 119909 (119905 minus 05) 119894 = 1119890minus1199051199092(119905minus05)119909 (119905 minus 05) 119894 = 2

(26)

119903(119905) is a right-continuousMarkov chain on the state spaceS =1 2 with the generator

119876 = [minus1 12 minus2] (27)

Theorem31 in [48] is void in determining the exponentialstability for such a system In fact taking 119881(119905 119909 119894) = |119909|2 and1198881 = 1198882 = 1 then by Itorsquos formula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le 2E(119909 (119905) minus 02119909 (119905 minus 05))sdot [( 1

16119905 cos 1199052 minus32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905)]+ E |119909 (119905 minus 05)|2 le ( 7

40119905 cos 1199052 minus34)E |119909 (119905)|2

+ ( 340119905 cos 1199052 minus

120)E |119909 (119905 minus 05)|2

(28)

6 Mathematical Problems in Engineering

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le 2E [(119909 (119905) minus 02119909 (119905 minus 05)) (18119905 cos 1199052 minus 1)

sdot 119909 (119905 minus 05)] + E |119909 (119905 minus 05)|2 le (18119905 cos 1199052 minus 1)

sdot E |119909 (119905)|2 + ( 110119905 cos 1199052 minus

25)E |119909 (119905 minus 05)|2

(29)

Obviously the conditions in Theorem 31 [48] do not holdBut we can judge the exponential stability for such system byusing our result Take 119881(119905 119909 119894) = |119909|2 and 1198881 = 1198882 = 1 By Itorsquosformula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le (18119905 cos 1199052 minus 3)E |119909 (119905) 119909 (119905 minus 05)|

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)| + E |119909 (119905 minus 05)|2

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)|

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18 (119902 + 1

2 ) 119905 cos 1199052E |119909 (119905)|2

= [18119905 cos 1199052 (119902 + 1) minus 3 (119902 + 1)2 + 119902]E |119909 (119905)|2

(30)

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le (18119905 cos 1199052 minus 1)E |119909 (119905)|2

+ (18119905 cos 1199052 minus 1)E |119909 (119905 minus 05)|2

+ E |119909 (119905 minus 05)|2

le [(14119905 cos 1199052 minus 2) 119902 + 12 + 119902]E |119909 (119905)|2

= (18119905 cos 1199052 (119902 + 1) + 119902)E |119909 (119905)|2

(31)

Choosing 119902 = 3 then EL119881(119905 119909(119905) 119909(119905 minus 05) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) where 120583(119905) = (12)119905 cos 1199052 minus 1 Here

Ｒ1

Ｒ2

minus015

minus01

minus005

0

005

01

015

10 20 30 40 500t

Figure 1 The state response of Example 1

120583(119905) is an unbounded function taking values in R so wecannot use any former results to judge the mean-squareexponential stability for such system But from our resultwe can determine its stability In fact taking 119879 = 1 thenint119905+1119905 120583(119904)119889119904 = minus12 lt 0 holds for all 119905 ge 0 Thus 119879 isinΩ120583 Additionally 119890120593120583 le 11989005 and 119902minus1119890120593120583(1 minus 119896)minus119901 le 3minus1 times11989005 times 08minus2 lt 1 By Theorem 7 the system is mean squareexponentially stable See Figure 1

5 Conclusion

In this paper we investigate the stability criterion for neu-tral stochastic delay differential equations with Markovianswitching By using stochastic analysis technique and Razu-mikhin approach we overcome the difficulty caused by theneutral item for the related system Finally we derive a newand novel Razumikhin exponential stability criterion Thefeature of the result is that the estimated upper bound forthe diffusion operator of Lyapunov function is allowed totake values on R even if it is allowed to be unboundedThe criterion can reduce some restrictiveness of the relatedresults that are existing in the previous literature An exampleis provided to show the superiority of the new exponentialstability criterion

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This work was supported by the China Postdoctoral ScienceFoundation (2018M632325)

Mathematical Problems in Engineering 7

References

[1] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 998ndash10182017

[2] Y Guo ldquoMean square exponential stability of stochastic delaycellular neural networksrdquo Electronic Journal of QualitativeTheory of Differential Equations no 34 pp 1ndash10 2013

[3] Q Zhu andHWang ldquoOutput feedback stabilization of stochas-tic feedforward systems with unknown control coefficients andunknown output functionrdquo Automatica vol 87 pp 166ndash1752018

[4] H Wang and Q Zhu ldquoFinite-time stabilization of high-orderstochastic nonlinear systems in strict-feedback formrdquoAutomat-ica vol 54 pp 284ndash291 2015

[5] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[6] Y Bai and X Mu ldquoGlobal asymptotic stability of a generalizedSIRS epidemic model with transfer from infectious to suscepti-blerdquo Journal of Applied Analysis and Computation vol 8 no 2pp 402ndash412 2018

[7] Q Zhu S Song andT Tang ldquoMean square exponential stabilityof stochastic nonlinear delay systemsrdquo International Journal ofControl vol 90 no 11 pp 2384ndash2393 2017

[8] Y Guo ldquoGlobal asymptotic stability analysis for integro-differential systems modeling neural networks with delaysrdquoZeitschrift fur Angewandte Mathematik und Physik vol 61 no6 pp 971ndash978 2010

[9] L Gao DWang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[10] Y Guo ldquoExponential stability analysis of travelling waves solu-tions for nonlinear delayed cellular neural networksrdquo Dynami-cal Systems vol 32 no 4 pp 490ndash503 2017

[11] Y Li Y Sun and FMeng ldquoNew criteria for exponential stabilityof switched time-varying systems with delays and nonlineardisturbancesrdquo Nonlinear Analysis Hybrid Systems vol 26 pp284ndash291 2017

[12] B Wang and Q Zhu ldquoStability analysis of Markov switchedstochastic differential equations with both stable and unstablesubsystemsrdquo Systems amp Control Letters vol 105 pp 55ndash61 2017

[13] L GWang K P Xu and QW Liu ldquoOn the stability of a mixedfunctional equation deriving fromadditive quadratic and cubicmappingsrdquo Acta Mathematica Sinica vol 30 no 6 pp 1033ndash1049 2014

[14] Q Zhu S Song and P Shi ldquoEffect of noise on the solutions ofnon-linear delay systemsrdquo IET Control Theory amp Applicationsvol 12 no 13 pp 1822ndash1829 2018

[15] G Liu S Xu Y Wei Z Qi and Z Zhang ldquoNew insightinto reachable set estimation for uncertain singular time-delaysystemsrdquo Applied Mathematics and Computation vol 320 pp769ndash780 2018

[16] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[17] Q Zhu ldquoAsymptotic stability in the pth moment for stochasticdifferential equations with Levy noiserdquo Journal of MathematicalAnalysis and Applications vol 416 no 1 pp 126ndash142 2014

[18] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[19] J Hu and A Xu ldquoOn stability of F-Gorenstein flat categoriesrdquoAlgebra Colloquium vol 23 no 2 pp 251ndash262 2016

[20] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[21] H Wang and Q Zhu ldquoGlobal Stabilization of StochasticNonlinear Systems via C 1 and C infin Controllersrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 11 pp 5880ndash5887 2017

[22] Y Guo ldquoMean square global asymptotic stability of stochasticrecurrent neural networks with distributed delaysrdquo AppliedMathematics andComputation vol 215 no 2 pp 791ndash795 2009

[23] Q Zhu ldquoStability analysis of stochastic delay differential equa-tions with Levy noiserdquo Systems amp Control Letters vol 118 pp62ndash68 2018

[24] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[25] W Qi Y Kao X Gao and Y Wei ldquoController design fortime-delay system with stochastic disturbance and actuatorsaturation via a new criterionrdquo Applied Mathematics and Com-putation vol 320 pp 535ndash546 2018

[26] L G Wang and B Liu ldquoFuzzy stability of a functional equationderiving from additive quadratic cubic and quartic functionsrdquoActa Mathematica Sinica vol 55 no 5 pp 841ndash854 2012

[27] Q Zhu and B Song ldquoExponential stability of impulsivenonlinear stochastic differential equations with mixed delaysrdquoNonlinear Analysis Real World Applications vol 12 no 5 pp2851ndash2860 2011

[28] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[29] Y-H Feng and C-M Liu ldquoStability of steady-state solutionsto Navier-Stokes-Poisson systemsrdquo Journal of MathematicalAnalysis and Applications vol 462 no 2 pp 1679ndash1694 2018

[30] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[31] T Li and Y F Yang ldquoStability of large steady-state solutionsto non-isentropic Euler-Maxwell systemsrdquo Journal of NanjingNormal University vol 40 no 4 pp 26ndash35 2017

[32] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquo Ukrainian Mathematical Journal vol 69 no 8 pp1049ndash1060 2017

[33] Q Zhu J Cao andR Rakkiyappan ldquoExponential input-to-statestability of stochastic Cohen-Grossberg neural networks withmixed delaysrdquoNonlinearDynamics vol 79 no 2 pp 1085ndash10982015

[34] L GWang andB Liu ldquoTheHyers-Ulam stability of a functionalequation deriving from quadratic and cubic functions in quasi-120573-normed spacesrdquo Acta Mathematica Sinica vol 26 no 12 pp2335ndash2348 2010

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

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Mathematical Problems in Engineering

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Mathematical Problems in Engineering 3

where

119881119909 (119905 119909 119894)= (120597119881 (119905 119909 119894)

1205971199091 120597119881 (119905 119909 119894)1205971199092 120597119881 (119905 119909 119894)

120597119909119899 )

119881119909119909 (119905 119909 119894) = (1205972119881(119905 119909 119894)120597119909119894120597119909119895 )

119899times119899

119881119905 (119905 119909 119894) = 120597119881 (119905 119909 119894)120597119905

(6)

In order to overcome the difficult caused by the neutralitem we need the following inequality

Lemma 3 Let 119901 ge 1 Then

(1 minus 120581)119901minus1 |119909|119901 minus 120581 (1 minus 120581)119901minus1 10038161003816100381610038161199101003816100381610038161003816119901 le 1003816100381610038161003816119909 minus 119906 (119905 119910 119894)1003816100381610038161003816119901le (1 + 120581)119901minus1 (|119909|119901 + 120581 10038161003816100381610038161199101003816100381610038161003816119901)

(7)

holds for any (119905 119909 119910 119894) isin R+ timesR119899 timesR119899 timesS

The following lemma which is important in the proof ofthe main result can be found in Lemma 1 of [62]

Lemma 4 Let 119879lowast be a constant Let a function 119908 [minus119879lowastinfin) 997888rarr [0infin) admit a sequence V119894 and positiveconstants V119886 and V119887 such that V0 = 1199050 V119894+1 minus V119894 isin [V119886 V119887] forall 119894 ge 0 119908 is continuous on [V119894 V119894+1) for all 119894 ge 0 and the leftlimit lim119905997888rarrVminus119894 119908(119905) exist Assume that there exists a constant120588 isin (0 1) such that 119908(119905) le 120588 sup119904isin[119905minus119879lowast 119905]119908(119904) holds for all119905 ge 1199050 Then

119908 (119905) le 119890((ln 120588)119879lowast)(119905minus1199050) sup119904isin[1199050minus119879lowast 1199050]

119908 (119904) (8)

holds for all 119905 ge 1199050The following definitions which are dated from [60] are

important in the proof of our main result

Definition 5 A piecewise continuous function 120583 is said to beaUSF if the following linear time-varying equation is globallyuniformly asymptotically stable

119889119910 (119905) = 120583 (119905) 119910 (119905) 119889119905 forall119905 ge 0 (9)

Definition 6 Let 120583(119905) be a USF Then the set

Ω120583 = 119879 gt 0 sup119905ge0

int119905+119879119905

120583 (119904) 119889119904 lt 0 (10)

is said to be the uniform convergence set of 120583(119905) For any 119879 ge0 the overshoot 120593120583(119879) fl 120593120583 of 120583(119905) is defined as follows

120593120583 = sup119905ge0

max120579isin[0119879]

int119905+120579119905

120583 (119904) 119889119904 (11)

3 Main Results

In this section we will use stochastic analysis theory Lya-punovrsquos method and Lemmas 3 and 4 to obtain a criterionfor 119901th moment exponential stability of system (2) Ourmainresult is the following

Theorem 7 Let 119901 1198881 1198882 120581 120591 be all positive numbers and 120583(119905)is a USF If there exist a function 119881 isin Ψ and a constant 119879 isin Ω120583such that the following conditions hold for all 119905 ge 0

(1) for all 119909 isin R119899 119894 isin S

1198881 |119909|119901 le 119881 (119905 119909 119894) le 1198882 |119909|119901 (12)

(2) the following Razumikhin-type condition holds

EL119881(119905 119909 (119905) 119909 (119905 minus 120591) 119903 (119905))le 120583 (119905)E119881 (119905 119909 (119905) 119903 (119905))

if min119894isin119878

E119881(119905 + 120579 119909 (119905 + 120579) 119894)le 119902max119894isin119878

E119881(119905 119909 (119905) 119894)

forall120579 isin [minus120591 0] where 119902 gt 11988821198881 119890120593120583 (1 minus 120581)minus119901

(13)

Then system (2) is 119901th moment exponentially stable here119909(119905) ≜ 119909(119905) minus 119906(119905 119909(119905 minus 120591) 119903(119905))Proof First of all we will prove the following fact if119902minus1E119881(119905 + 120579 119909(119905 + 120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905)) holds for all120579 isin [minus120591 0] which implies EL119881(119905 119909(119905) 119909(119905 minus 120591) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) then for any 119905 gt 119879E119881 (119905 119909 (119905) 119903 (119905))

le maxE119881 (119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879)) 119890int119905119905minus119879 120583(119904)119889119904

119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881 (119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583

(14)

In order to prove inequality (14) we consider the followingtwo cases

(A) sup120579isin[minus1205910]E119881(119904 + 120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904)119903(119904)) holds for all 119904 isin [119905 minus 119879 119905](B) sup120579isin[minus1205910]E119881(119904 + 120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904)119903(119904)) does not hold for some 119904 isin [119905 minus 119879 119905]For (A) we know that EL119881(119904 119909(119904) 119909(119904 minus 120591) 119903(119904)) le120583(119904)E119881(119904 119909(119904) 119903(119904)) holds for any 119904 isin [119905 minus 119879 119905] By using

4 Mathematical Problems in Engineering

Itorsquos formula and the standard stopping times technique weobtain that

E(119890minusint119905119905minus119879 120583(119904)119889119904119881(119905 119909 (119905) 119903 (119905))) = E119881(119905 minus 119879 119909 (119905minus 119879) 119903 (119905 minus 119879))+ Eint119905119905minus119879

[minus120583 (119904) 119890minus int119904119905minus119879 120583(V)119889V119881 (119904 119909 (119904) 119903 (119904))+ 119890minus int119904119905minus119879 120583(V)119889VL119881(119904 119909 (119904) 119909 (119904 minus 120591) 119903 (119904))] 119889119904le E119881(119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879))

(15)

which implies that E119881(119905 119909(119905) 119903(119905)) le E119881(119905minus119879 119909(119905 minus119879) 119903(119905 minus119879))119890int119905119905minus119879 120583(119904)119889119904

For (B) define 119905lowast = sup119904 isin [119905 minus 119879 119905] sup120579isin[minus1205910]E119881(119904 +120579 119909(119904 + 120579) 119903(119904 + 120579)) gt 119902E119881(119904 119909(119904) 119903(119904)) Then 119905lowast lt 119905 or119905lowast = 119905 If 119905lowast lt 119905 then for all 119904 isin [119905lowast 119905] sup120579isin[minus1205910]E119881(119904 +120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904) 119903(119904)) So according to theresult of (A) we obtain that

E119881(119905 119909 (119905) 119903 (119905)) le 119890int119905119905lowast 120583(119904)119889119904E119881(119905lowast 119909 (119905lowast) 119903 (119905lowast))= 119890int119905119905lowast 120583(119904)119889119904119902minus1 sup

120579isin[minus1205910]E119881(119905lowast

+ 120579 119909 (119905lowast + 120579) 119903 (119905lowast + 120579))le 119890int119905119905lowast 120583(119904)119889119904119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904+ 120579 119909 (119904 + 120579) 119903 (119904 + 120579))le 119890120593120583119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904+ 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

(16)

If 119905lowast = 119905 from the definition of 119905lowast we haveE119881(119905 119909 (119905) 119903 (119905)) = E119881(119905lowast 119909 (119905lowast) 119903 (119905lowast))le 119902minus1 sup120579isin[minus1205910]

E119881 (119905lowast + 120579 119909 (119905lowast + 120579) 119903 (119905lowast + 120579))le 119890120593120583119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579)) (17)

Combining the above three inequalities we can see thatinequality (14) holds

Assume that 119902minus1E119881(119905+120579 119909(119905+120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905))holds for all 120579 isin [minus120591 0] then 119902minus1min119894isin119878E119881(119905 + 120579 119909(119905 +120579) 119894) le 119902minus1E119881(119905 + 120579 119909(119905 + 120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905)) lemax119894isin119878E119881(119905 119909(119905) 119894) holds for all 120579 isin [minus120591 0] According tocondition (2) we know that EL119881(119905 119909(119905) 119909(119905 minus 120591) 119903(119905)) le

120583(119905)E119881(119905 119909(119905) 119903(119905))Thus from inequality (14) condition (1)and Lemma 3 we conclude that for any 119905 gt 119879E119881(119905 119909 (119905) 119903 (119905))

le maxE119881(119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879)) 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max1198882E |119909 (119905 minus 119879)|119901 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max1198882 (1 + 120581)119901minus1

sdot (E |119909 (119905 minus 119879)|119901 + 120581E |119909 (119905 minus 119879 minus 120591)|119901) 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579119909 (119904 + 120579) 119903 (119904 + 120579))

(18)

By condition (1) we can see that

E |119909 (119905)|119901 le 11988821198881max11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

fl 120578 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 (19)

Using Lemma 3 again we obtain that

E |119909 (119905)|119901le [E |119909 (119905)|119901 + 120581 (1 minus 120581)119901minus1 E |119909 (119905 minus 120591)|119901] (1 minus 120581)1minus119901= (1 minus 120581)1minus119901 E |119909 (119905)|119901 + 120581E |119909 (119905 minus 120591)|119901le [(1 minus 120581)1minus119901 120578 + 120581] sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 (20)

In order to use Lemma 4 we need to justify that (1minus120581)1minus119901120578+120581 lt 1 In fact on one hand due to 119902 gt (11988821198881)119890120593120583(1 minus 120581)minus119901there exists a constant 120588 isin (0 1) such that (11988821198881)119902minus1119890120593120583 (1 minus120581)1minus119901 le 120588 minus 120581 On the other hand due to 120583(119905) being a USFand 119879 isin Ω120583 there exists a constant 1205881 isin (0 1) such that

Mathematical Problems in Engineering 5

119890int119905119905minus119879 120583(119904)119889119904 lt (11988811198882)2((1 minus 120581)(1 + 120581))119901((1205881 minus 120581)(1 minus 120581)) lt 1Thus

E |119909 (119905)|119901 le [(1 minus 120581)1minus119901 120578 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 = [(1 minus 120581)1minus119901 11988821198881sdotmax11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le [max 1205881 minus 120581 120588 minus 120581 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879minus120591119905]

E |119909 (119904)|119901

(21)

where 120588lowast = 120588 or 1205881 isin (0 1) By Lemma 4 we derive that

E |119909 (119905)|119901 le 119890((ln 120588lowast)(119879+120591))(119905minus119879) sup119904isin[minus120591119879]

E |119909 (119904)|119901 (22)

which implies that

lim sup119905997888rarrinfin

lnE |119909 (119905)|119901119905 = ln 120588lowast

119879 + 120591 lt 0 (23)

In other words we have proved that system (2) is119901thmomentexponentially stable

Remark 8 Here by using stochastic analysis theory somenotions as uniformly stable function (USF) overshoot ofthe USF that dated from [60 61] and some inequalitytechniques we obtain a novel exponential stability criterionwith respect to the related systems The method we use hereis rather different from the traditional Razumikhin approachThus we provide a new method to investigate the stabilityanalysis for neutral stochastic delayed differential equationswith Markovian switchings In the future we can developthis approach to study the stability for impulsive neutralstochastic delayed differential equations

Remark 9 If system (2) has no Markovian switching and120583(119905) equiv 120583 lt 0 then 119890120593120583 = the condition of Theorem 7 119902 gt(11988821198881)119890120593120583(1 minus 120581)minus119901 becomes 119902 gt (11988821198881)(1 minus 120581)minus119901 which is thesame as the conditions in [46 47] If the model we consideredhas no Markovian switching stochastic disturbance andneutral item then we can obtain the same result as Theorem1 in [60] Thus our result is the generalization of Theorem 1in [60] and the main results in [46 47]

Remark 10 In Theorem 31 of [53] the condition (33)ensured that the coefficient of the estimated upper boundfor the diffusion operator of Lyapunov function is a negative

value and the condition of Theorem 31 in [48] requiredthat the coefficient function of the estimated upper boundfor the diffusion operator of Lyapunov function is a negativedefinite function which are very conservative in the practiceif the system is time-varying system But our conditions allowthis function to take values on R and even allow it to beunbounded Thus our result can be used more widely and isless conservative than the existing results See Example 1 formore details

4 An Example

In this section we will give an example to illustrate thevalidity and significance of our result

Example 1 Now we will consider the following neutral-type stochastic delay differential system with Markovianswitching

119889 [119909 (119905) minus 02119909 (119905 minus 05)]= 119891 (119905 119909 (119905) 119909 (t minus 05) 119903 (119905)) 119889119905

+ 119892 (119905 119909 (119905) 119909 (119905 minus 05) 119903 (119905)) 119889119861119905(24)

where119891 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

=

( 116119905 cos 1199052 minus

32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905) 119894 = 1(18119905 cos 1199052 minus 1) 119909 (119905 minus 05) 119894 = 2

(25)

and119892 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

= (tanh 119905) 119909 (119905 minus 05) 119894 = 1119890minus1199051199092(119905minus05)119909 (119905 minus 05) 119894 = 2

(26)

119903(119905) is a right-continuousMarkov chain on the state spaceS =1 2 with the generator

119876 = [minus1 12 minus2] (27)

Theorem31 in [48] is void in determining the exponentialstability for such a system In fact taking 119881(119905 119909 119894) = |119909|2 and1198881 = 1198882 = 1 then by Itorsquos formula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le 2E(119909 (119905) minus 02119909 (119905 minus 05))sdot [( 1

16119905 cos 1199052 minus32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905)]+ E |119909 (119905 minus 05)|2 le ( 7

40119905 cos 1199052 minus34)E |119909 (119905)|2

+ ( 340119905 cos 1199052 minus

120)E |119909 (119905 minus 05)|2

(28)

6 Mathematical Problems in Engineering

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le 2E [(119909 (119905) minus 02119909 (119905 minus 05)) (18119905 cos 1199052 minus 1)

sdot 119909 (119905 minus 05)] + E |119909 (119905 minus 05)|2 le (18119905 cos 1199052 minus 1)

sdot E |119909 (119905)|2 + ( 110119905 cos 1199052 minus

25)E |119909 (119905 minus 05)|2

(29)

Obviously the conditions in Theorem 31 [48] do not holdBut we can judge the exponential stability for such system byusing our result Take 119881(119905 119909 119894) = |119909|2 and 1198881 = 1198882 = 1 By Itorsquosformula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le (18119905 cos 1199052 minus 3)E |119909 (119905) 119909 (119905 minus 05)|

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)| + E |119909 (119905 minus 05)|2

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)|

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18 (119902 + 1

2 ) 119905 cos 1199052E |119909 (119905)|2

= [18119905 cos 1199052 (119902 + 1) minus 3 (119902 + 1)2 + 119902]E |119909 (119905)|2

(30)

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le (18119905 cos 1199052 minus 1)E |119909 (119905)|2

+ (18119905 cos 1199052 minus 1)E |119909 (119905 minus 05)|2

+ E |119909 (119905 minus 05)|2

le [(14119905 cos 1199052 minus 2) 119902 + 12 + 119902]E |119909 (119905)|2

= (18119905 cos 1199052 (119902 + 1) + 119902)E |119909 (119905)|2

(31)

Choosing 119902 = 3 then EL119881(119905 119909(119905) 119909(119905 minus 05) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) where 120583(119905) = (12)119905 cos 1199052 minus 1 Here

Ｒ1

Ｒ2

minus015

minus01

minus005

0

005

01

015

10 20 30 40 500t

Figure 1 The state response of Example 1

120583(119905) is an unbounded function taking values in R so wecannot use any former results to judge the mean-squareexponential stability for such system But from our resultwe can determine its stability In fact taking 119879 = 1 thenint119905+1119905 120583(119904)119889119904 = minus12 lt 0 holds for all 119905 ge 0 Thus 119879 isinΩ120583 Additionally 119890120593120583 le 11989005 and 119902minus1119890120593120583(1 minus 119896)minus119901 le 3minus1 times11989005 times 08minus2 lt 1 By Theorem 7 the system is mean squareexponentially stable See Figure 1

5 Conclusion

In this paper we investigate the stability criterion for neu-tral stochastic delay differential equations with Markovianswitching By using stochastic analysis technique and Razu-mikhin approach we overcome the difficulty caused by theneutral item for the related system Finally we derive a newand novel Razumikhin exponential stability criterion Thefeature of the result is that the estimated upper bound forthe diffusion operator of Lyapunov function is allowed totake values on R even if it is allowed to be unboundedThe criterion can reduce some restrictiveness of the relatedresults that are existing in the previous literature An exampleis provided to show the superiority of the new exponentialstability criterion

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This work was supported by the China Postdoctoral ScienceFoundation (2018M632325)

Mathematical Problems in Engineering 7

References

[1] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 998ndash10182017

[2] Y Guo ldquoMean square exponential stability of stochastic delaycellular neural networksrdquo Electronic Journal of QualitativeTheory of Differential Equations no 34 pp 1ndash10 2013

[3] Q Zhu andHWang ldquoOutput feedback stabilization of stochas-tic feedforward systems with unknown control coefficients andunknown output functionrdquo Automatica vol 87 pp 166ndash1752018

[4] H Wang and Q Zhu ldquoFinite-time stabilization of high-orderstochastic nonlinear systems in strict-feedback formrdquoAutomat-ica vol 54 pp 284ndash291 2015

[5] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[6] Y Bai and X Mu ldquoGlobal asymptotic stability of a generalizedSIRS epidemic model with transfer from infectious to suscepti-blerdquo Journal of Applied Analysis and Computation vol 8 no 2pp 402ndash412 2018

[7] Q Zhu S Song andT Tang ldquoMean square exponential stabilityof stochastic nonlinear delay systemsrdquo International Journal ofControl vol 90 no 11 pp 2384ndash2393 2017

[8] Y Guo ldquoGlobal asymptotic stability analysis for integro-differential systems modeling neural networks with delaysrdquoZeitschrift fur Angewandte Mathematik und Physik vol 61 no6 pp 971ndash978 2010

[9] L Gao DWang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[10] Y Guo ldquoExponential stability analysis of travelling waves solu-tions for nonlinear delayed cellular neural networksrdquo Dynami-cal Systems vol 32 no 4 pp 490ndash503 2017

[11] Y Li Y Sun and FMeng ldquoNew criteria for exponential stabilityof switched time-varying systems with delays and nonlineardisturbancesrdquo Nonlinear Analysis Hybrid Systems vol 26 pp284ndash291 2017

[12] B Wang and Q Zhu ldquoStability analysis of Markov switchedstochastic differential equations with both stable and unstablesubsystemsrdquo Systems amp Control Letters vol 105 pp 55ndash61 2017

[13] L GWang K P Xu and QW Liu ldquoOn the stability of a mixedfunctional equation deriving fromadditive quadratic and cubicmappingsrdquo Acta Mathematica Sinica vol 30 no 6 pp 1033ndash1049 2014

[14] Q Zhu S Song and P Shi ldquoEffect of noise on the solutions ofnon-linear delay systemsrdquo IET Control Theory amp Applicationsvol 12 no 13 pp 1822ndash1829 2018

[15] G Liu S Xu Y Wei Z Qi and Z Zhang ldquoNew insightinto reachable set estimation for uncertain singular time-delaysystemsrdquo Applied Mathematics and Computation vol 320 pp769ndash780 2018

[16] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[17] Q Zhu ldquoAsymptotic stability in the pth moment for stochasticdifferential equations with Levy noiserdquo Journal of MathematicalAnalysis and Applications vol 416 no 1 pp 126ndash142 2014

[18] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[19] J Hu and A Xu ldquoOn stability of F-Gorenstein flat categoriesrdquoAlgebra Colloquium vol 23 no 2 pp 251ndash262 2016

[20] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[21] H Wang and Q Zhu ldquoGlobal Stabilization of StochasticNonlinear Systems via C 1 and C infin Controllersrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 11 pp 5880ndash5887 2017

[22] Y Guo ldquoMean square global asymptotic stability of stochasticrecurrent neural networks with distributed delaysrdquo AppliedMathematics andComputation vol 215 no 2 pp 791ndash795 2009

[23] Q Zhu ldquoStability analysis of stochastic delay differential equa-tions with Levy noiserdquo Systems amp Control Letters vol 118 pp62ndash68 2018

[24] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[25] W Qi Y Kao X Gao and Y Wei ldquoController design fortime-delay system with stochastic disturbance and actuatorsaturation via a new criterionrdquo Applied Mathematics and Com-putation vol 320 pp 535ndash546 2018

[26] L G Wang and B Liu ldquoFuzzy stability of a functional equationderiving from additive quadratic cubic and quartic functionsrdquoActa Mathematica Sinica vol 55 no 5 pp 841ndash854 2012

[27] Q Zhu and B Song ldquoExponential stability of impulsivenonlinear stochastic differential equations with mixed delaysrdquoNonlinear Analysis Real World Applications vol 12 no 5 pp2851ndash2860 2011

[28] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[29] Y-H Feng and C-M Liu ldquoStability of steady-state solutionsto Navier-Stokes-Poisson systemsrdquo Journal of MathematicalAnalysis and Applications vol 462 no 2 pp 1679ndash1694 2018

[30] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[31] T Li and Y F Yang ldquoStability of large steady-state solutionsto non-isentropic Euler-Maxwell systemsrdquo Journal of NanjingNormal University vol 40 no 4 pp 26ndash35 2017

[32] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquo Ukrainian Mathematical Journal vol 69 no 8 pp1049ndash1060 2017

[33] Q Zhu J Cao andR Rakkiyappan ldquoExponential input-to-statestability of stochastic Cohen-Grossberg neural networks withmixed delaysrdquoNonlinearDynamics vol 79 no 2 pp 1085ndash10982015

[34] L GWang andB Liu ldquoTheHyers-Ulam stability of a functionalequation deriving from quadratic and cubic functions in quasi-120573-normed spacesrdquo Acta Mathematica Sinica vol 26 no 12 pp2335ndash2348 2010

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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4 Mathematical Problems in Engineering

Itorsquos formula and the standard stopping times technique weobtain that

E(119890minusint119905119905minus119879 120583(119904)119889119904119881(119905 119909 (119905) 119903 (119905))) = E119881(119905 minus 119879 119909 (119905minus 119879) 119903 (119905 minus 119879))+ Eint119905119905minus119879

[minus120583 (119904) 119890minus int119904119905minus119879 120583(V)119889V119881 (119904 119909 (119904) 119903 (119904))+ 119890minus int119904119905minus119879 120583(V)119889VL119881(119904 119909 (119904) 119909 (119904 minus 120591) 119903 (119904))] 119889119904le E119881(119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879))

(15)

which implies that E119881(119905 119909(119905) 119903(119905)) le E119881(119905minus119879 119909(119905 minus119879) 119903(119905 minus119879))119890int119905119905minus119879 120583(119904)119889119904

For (B) define 119905lowast = sup119904 isin [119905 minus 119879 119905] sup120579isin[minus1205910]E119881(119904 +120579 119909(119904 + 120579) 119903(119904 + 120579)) gt 119902E119881(119904 119909(119904) 119903(119904)) Then 119905lowast lt 119905 or119905lowast = 119905 If 119905lowast lt 119905 then for all 119904 isin [119905lowast 119905] sup120579isin[minus1205910]E119881(119904 +120579 119909(119904 + 120579) 119903(119904 + 120579)) le 119902E119881(119904 119909(119904) 119903(119904)) So according to theresult of (A) we obtain that

E119881(119905 119909 (119905) 119903 (119905)) le 119890int119905119905lowast 120583(119904)119889119904E119881(119905lowast 119909 (119905lowast) 119903 (119905lowast))= 119890int119905119905lowast 120583(119904)119889119904119902minus1 sup

120579isin[minus1205910]E119881(119905lowast

+ 120579 119909 (119905lowast + 120579) 119903 (119905lowast + 120579))le 119890int119905119905lowast 120583(119904)119889119904119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904+ 120579 119909 (119904 + 120579) 119903 (119904 + 120579))le 119890120593120583119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904+ 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

(16)

If 119905lowast = 119905 from the definition of 119905lowast we haveE119881(119905 119909 (119905) 119903 (119905)) = E119881(119905lowast 119909 (119905lowast) 119903 (119905lowast))le 119902minus1 sup120579isin[minus1205910]

E119881 (119905lowast + 120579 119909 (119905lowast + 120579) 119903 (119905lowast + 120579))le 119890120593120583119902minus1 sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579)) (17)

Combining the above three inequalities we can see thatinequality (14) holds

Assume that 119902minus1E119881(119905+120579 119909(119905+120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905))holds for all 120579 isin [minus120591 0] then 119902minus1min119894isin119878E119881(119905 + 120579 119909(119905 +120579) 119894) le 119902minus1E119881(119905 + 120579 119909(119905 + 120579) 119903(119905)) le E119881(119905 119909(119905) 119903(119905)) lemax119894isin119878E119881(119905 119909(119905) 119894) holds for all 120579 isin [minus120591 0] According tocondition (2) we know that EL119881(119905 119909(119905) 119909(119905 minus 120591) 119903(119905)) le

120583(119905)E119881(119905 119909(119905) 119903(119905))Thus from inequality (14) condition (1)and Lemma 3 we conclude that for any 119905 gt 119879E119881(119905 119909 (119905) 119903 (119905))

le maxE119881(119905 minus 119879 119909 (119905 minus 119879) 119903 (119905 minus 119879)) 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max1198882E |119909 (119905 minus 119879)|119901 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max1198882 (1 + 120581)119901minus1

sdot (E |119909 (119905 minus 119879)|119901 + 120581E |119909 (119905 minus 119879 minus 120591)|119901) 119890int119905119905minus119879 120583(119904)119889119904119902minus1 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579 119909 (119904 + 120579) 119903 (119904 + 120579))

sdot 119890120593120583 le max11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E119881(119904 + 120579119909 (119904 + 120579) 119903 (119904 + 120579))

(18)

By condition (1) we can see that

E |119909 (119905)|119901 le 11988821198881max11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

fl 120578 sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 (19)

Using Lemma 3 again we obtain that

E |119909 (119905)|119901le [E |119909 (119905)|119901 + 120581 (1 minus 120581)119901minus1 E |119909 (119905 minus 120591)|119901] (1 minus 120581)1minus119901= (1 minus 120581)1minus119901 E |119909 (119905)|119901 + 120581E |119909 (119905 minus 120591)|119901le [(1 minus 120581)1minus119901 120578 + 120581] sup

119904isin[119905minus119879119905]sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 (20)

In order to use Lemma 4 we need to justify that (1minus120581)1minus119901120578+120581 lt 1 In fact on one hand due to 119902 gt (11988821198881)119890120593120583(1 minus 120581)minus119901there exists a constant 120588 isin (0 1) such that (11988821198881)119902minus1119890120593120583 (1 minus120581)1minus119901 le 120588 minus 120581 On the other hand due to 120583(119905) being a USFand 119879 isin Ω120583 there exists a constant 1205881 isin (0 1) such that

Mathematical Problems in Engineering 5

119890int119905119905minus119879 120583(119904)119889119904 lt (11988811198882)2((1 minus 120581)(1 + 120581))119901((1205881 minus 120581)(1 minus 120581)) lt 1Thus

E |119909 (119905)|119901 le [(1 minus 120581)1minus119901 120578 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 = [(1 minus 120581)1minus119901 11988821198881sdotmax11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le [max 1205881 minus 120581 120588 minus 120581 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879minus120591119905]

E |119909 (119904)|119901

(21)

where 120588lowast = 120588 or 1205881 isin (0 1) By Lemma 4 we derive that

E |119909 (119905)|119901 le 119890((ln 120588lowast)(119879+120591))(119905minus119879) sup119904isin[minus120591119879]

E |119909 (119904)|119901 (22)

which implies that

lim sup119905997888rarrinfin

lnE |119909 (119905)|119901119905 = ln 120588lowast

119879 + 120591 lt 0 (23)

In other words we have proved that system (2) is119901thmomentexponentially stable

Remark 8 Here by using stochastic analysis theory somenotions as uniformly stable function (USF) overshoot ofthe USF that dated from [60 61] and some inequalitytechniques we obtain a novel exponential stability criterionwith respect to the related systems The method we use hereis rather different from the traditional Razumikhin approachThus we provide a new method to investigate the stabilityanalysis for neutral stochastic delayed differential equationswith Markovian switchings In the future we can developthis approach to study the stability for impulsive neutralstochastic delayed differential equations

Remark 9 If system (2) has no Markovian switching and120583(119905) equiv 120583 lt 0 then 119890120593120583 = the condition of Theorem 7 119902 gt(11988821198881)119890120593120583(1 minus 120581)minus119901 becomes 119902 gt (11988821198881)(1 minus 120581)minus119901 which is thesame as the conditions in [46 47] If the model we consideredhas no Markovian switching stochastic disturbance andneutral item then we can obtain the same result as Theorem1 in [60] Thus our result is the generalization of Theorem 1in [60] and the main results in [46 47]

Remark 10 In Theorem 31 of [53] the condition (33)ensured that the coefficient of the estimated upper boundfor the diffusion operator of Lyapunov function is a negative

value and the condition of Theorem 31 in [48] requiredthat the coefficient function of the estimated upper boundfor the diffusion operator of Lyapunov function is a negativedefinite function which are very conservative in the practiceif the system is time-varying system But our conditions allowthis function to take values on R and even allow it to beunbounded Thus our result can be used more widely and isless conservative than the existing results See Example 1 formore details

4 An Example

In this section we will give an example to illustrate thevalidity and significance of our result

Example 1 Now we will consider the following neutral-type stochastic delay differential system with Markovianswitching

119889 [119909 (119905) minus 02119909 (119905 minus 05)]= 119891 (119905 119909 (119905) 119909 (t minus 05) 119903 (119905)) 119889119905

+ 119892 (119905 119909 (119905) 119909 (119905 minus 05) 119903 (119905)) 119889119861119905(24)

where119891 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

=

( 116119905 cos 1199052 minus

32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905) 119894 = 1(18119905 cos 1199052 minus 1) 119909 (119905 minus 05) 119894 = 2

(25)

and119892 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

= (tanh 119905) 119909 (119905 minus 05) 119894 = 1119890minus1199051199092(119905minus05)119909 (119905 minus 05) 119894 = 2

(26)

119903(119905) is a right-continuousMarkov chain on the state spaceS =1 2 with the generator

119876 = [minus1 12 minus2] (27)

Theorem31 in [48] is void in determining the exponentialstability for such a system In fact taking 119881(119905 119909 119894) = |119909|2 and1198881 = 1198882 = 1 then by Itorsquos formula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le 2E(119909 (119905) minus 02119909 (119905 minus 05))sdot [( 1

16119905 cos 1199052 minus32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905)]+ E |119909 (119905 minus 05)|2 le ( 7

40119905 cos 1199052 minus34)E |119909 (119905)|2

+ ( 340119905 cos 1199052 minus

120)E |119909 (119905 minus 05)|2

(28)

6 Mathematical Problems in Engineering

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le 2E [(119909 (119905) minus 02119909 (119905 minus 05)) (18119905 cos 1199052 minus 1)

sdot 119909 (119905 minus 05)] + E |119909 (119905 minus 05)|2 le (18119905 cos 1199052 minus 1)

sdot E |119909 (119905)|2 + ( 110119905 cos 1199052 minus

25)E |119909 (119905 minus 05)|2

(29)

Obviously the conditions in Theorem 31 [48] do not holdBut we can judge the exponential stability for such system byusing our result Take 119881(119905 119909 119894) = |119909|2 and 1198881 = 1198882 = 1 By Itorsquosformula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le (18119905 cos 1199052 minus 3)E |119909 (119905) 119909 (119905 minus 05)|

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)| + E |119909 (119905 minus 05)|2

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)|

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18 (119902 + 1

2 ) 119905 cos 1199052E |119909 (119905)|2

= [18119905 cos 1199052 (119902 + 1) minus 3 (119902 + 1)2 + 119902]E |119909 (119905)|2

(30)

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le (18119905 cos 1199052 minus 1)E |119909 (119905)|2

+ (18119905 cos 1199052 minus 1)E |119909 (119905 minus 05)|2

+ E |119909 (119905 minus 05)|2

le [(14119905 cos 1199052 minus 2) 119902 + 12 + 119902]E |119909 (119905)|2

= (18119905 cos 1199052 (119902 + 1) + 119902)E |119909 (119905)|2

(31)

Choosing 119902 = 3 then EL119881(119905 119909(119905) 119909(119905 minus 05) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) where 120583(119905) = (12)119905 cos 1199052 minus 1 Here

Ｒ1

Ｒ2

minus015

minus01

minus005

0

005

01

015

10 20 30 40 500t

Figure 1 The state response of Example 1

120583(119905) is an unbounded function taking values in R so wecannot use any former results to judge the mean-squareexponential stability for such system But from our resultwe can determine its stability In fact taking 119879 = 1 thenint119905+1119905 120583(119904)119889119904 = minus12 lt 0 holds for all 119905 ge 0 Thus 119879 isinΩ120583 Additionally 119890120593120583 le 11989005 and 119902minus1119890120593120583(1 minus 119896)minus119901 le 3minus1 times11989005 times 08minus2 lt 1 By Theorem 7 the system is mean squareexponentially stable See Figure 1

5 Conclusion

In this paper we investigate the stability criterion for neu-tral stochastic delay differential equations with Markovianswitching By using stochastic analysis technique and Razu-mikhin approach we overcome the difficulty caused by theneutral item for the related system Finally we derive a newand novel Razumikhin exponential stability criterion Thefeature of the result is that the estimated upper bound forthe diffusion operator of Lyapunov function is allowed totake values on R even if it is allowed to be unboundedThe criterion can reduce some restrictiveness of the relatedresults that are existing in the previous literature An exampleis provided to show the superiority of the new exponentialstability criterion

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This work was supported by the China Postdoctoral ScienceFoundation (2018M632325)

Mathematical Problems in Engineering 7

References

[1] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 998ndash10182017

[2] Y Guo ldquoMean square exponential stability of stochastic delaycellular neural networksrdquo Electronic Journal of QualitativeTheory of Differential Equations no 34 pp 1ndash10 2013

[3] Q Zhu andHWang ldquoOutput feedback stabilization of stochas-tic feedforward systems with unknown control coefficients andunknown output functionrdquo Automatica vol 87 pp 166ndash1752018

[4] H Wang and Q Zhu ldquoFinite-time stabilization of high-orderstochastic nonlinear systems in strict-feedback formrdquoAutomat-ica vol 54 pp 284ndash291 2015

[5] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[6] Y Bai and X Mu ldquoGlobal asymptotic stability of a generalizedSIRS epidemic model with transfer from infectious to suscepti-blerdquo Journal of Applied Analysis and Computation vol 8 no 2pp 402ndash412 2018

[7] Q Zhu S Song andT Tang ldquoMean square exponential stabilityof stochastic nonlinear delay systemsrdquo International Journal ofControl vol 90 no 11 pp 2384ndash2393 2017

[8] Y Guo ldquoGlobal asymptotic stability analysis for integro-differential systems modeling neural networks with delaysrdquoZeitschrift fur Angewandte Mathematik und Physik vol 61 no6 pp 971ndash978 2010

[9] L Gao DWang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[10] Y Guo ldquoExponential stability analysis of travelling waves solu-tions for nonlinear delayed cellular neural networksrdquo Dynami-cal Systems vol 32 no 4 pp 490ndash503 2017

[11] Y Li Y Sun and FMeng ldquoNew criteria for exponential stabilityof switched time-varying systems with delays and nonlineardisturbancesrdquo Nonlinear Analysis Hybrid Systems vol 26 pp284ndash291 2017

[12] B Wang and Q Zhu ldquoStability analysis of Markov switchedstochastic differential equations with both stable and unstablesubsystemsrdquo Systems amp Control Letters vol 105 pp 55ndash61 2017

[13] L GWang K P Xu and QW Liu ldquoOn the stability of a mixedfunctional equation deriving fromadditive quadratic and cubicmappingsrdquo Acta Mathematica Sinica vol 30 no 6 pp 1033ndash1049 2014

[14] Q Zhu S Song and P Shi ldquoEffect of noise on the solutions ofnon-linear delay systemsrdquo IET Control Theory amp Applicationsvol 12 no 13 pp 1822ndash1829 2018

[15] G Liu S Xu Y Wei Z Qi and Z Zhang ldquoNew insightinto reachable set estimation for uncertain singular time-delaysystemsrdquo Applied Mathematics and Computation vol 320 pp769ndash780 2018

[16] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[17] Q Zhu ldquoAsymptotic stability in the pth moment for stochasticdifferential equations with Levy noiserdquo Journal of MathematicalAnalysis and Applications vol 416 no 1 pp 126ndash142 2014

[18] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[19] J Hu and A Xu ldquoOn stability of F-Gorenstein flat categoriesrdquoAlgebra Colloquium vol 23 no 2 pp 251ndash262 2016

[20] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[21] H Wang and Q Zhu ldquoGlobal Stabilization of StochasticNonlinear Systems via C 1 and C infin Controllersrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 11 pp 5880ndash5887 2017

[22] Y Guo ldquoMean square global asymptotic stability of stochasticrecurrent neural networks with distributed delaysrdquo AppliedMathematics andComputation vol 215 no 2 pp 791ndash795 2009

[23] Q Zhu ldquoStability analysis of stochastic delay differential equa-tions with Levy noiserdquo Systems amp Control Letters vol 118 pp62ndash68 2018

[24] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[25] W Qi Y Kao X Gao and Y Wei ldquoController design fortime-delay system with stochastic disturbance and actuatorsaturation via a new criterionrdquo Applied Mathematics and Com-putation vol 320 pp 535ndash546 2018

[26] L G Wang and B Liu ldquoFuzzy stability of a functional equationderiving from additive quadratic cubic and quartic functionsrdquoActa Mathematica Sinica vol 55 no 5 pp 841ndash854 2012

[27] Q Zhu and B Song ldquoExponential stability of impulsivenonlinear stochastic differential equations with mixed delaysrdquoNonlinear Analysis Real World Applications vol 12 no 5 pp2851ndash2860 2011

[28] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[29] Y-H Feng and C-M Liu ldquoStability of steady-state solutionsto Navier-Stokes-Poisson systemsrdquo Journal of MathematicalAnalysis and Applications vol 462 no 2 pp 1679ndash1694 2018

[30] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[31] T Li and Y F Yang ldquoStability of large steady-state solutionsto non-isentropic Euler-Maxwell systemsrdquo Journal of NanjingNormal University vol 40 no 4 pp 26ndash35 2017

[32] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquo Ukrainian Mathematical Journal vol 69 no 8 pp1049ndash1060 2017

[33] Q Zhu J Cao andR Rakkiyappan ldquoExponential input-to-statestability of stochastic Cohen-Grossberg neural networks withmixed delaysrdquoNonlinearDynamics vol 79 no 2 pp 1085ndash10982015

[34] L GWang andB Liu ldquoTheHyers-Ulam stability of a functionalequation deriving from quadratic and cubic functions in quasi-120573-normed spacesrdquo Acta Mathematica Sinica vol 26 no 12 pp2335ndash2348 2010

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

119890int119905119905minus119879 120583(119904)119889119904 lt (11988811198882)2((1 minus 120581)(1 + 120581))119901((1205881 minus 120581)(1 minus 120581)) lt 1Thus

E |119909 (119905)|119901 le [(1 minus 120581)1minus119901 120578 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901 = [(1 minus 120581)1minus119901 11988821198881sdotmax11988821198881 (1 + 120581)119901 119890int119905119905minus119879 120583(119904)119889119904 119902minus1119890120593120583 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le [max 1205881 minus 120581 120588 minus 120581 + 120581]sdot sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879119905]

sup120579isin[minus1205910]

E |119909 (119904 + 120579)|119901

le 120588lowast sup119904isin[119905minus119879minus120591119905]

E |119909 (119904)|119901

(21)

where 120588lowast = 120588 or 1205881 isin (0 1) By Lemma 4 we derive that

E |119909 (119905)|119901 le 119890((ln 120588lowast)(119879+120591))(119905minus119879) sup119904isin[minus120591119879]

E |119909 (119904)|119901 (22)

which implies that

lim sup119905997888rarrinfin

lnE |119909 (119905)|119901119905 = ln 120588lowast

119879 + 120591 lt 0 (23)

In other words we have proved that system (2) is119901thmomentexponentially stable

Remark 8 Here by using stochastic analysis theory somenotions as uniformly stable function (USF) overshoot ofthe USF that dated from [60 61] and some inequalitytechniques we obtain a novel exponential stability criterionwith respect to the related systems The method we use hereis rather different from the traditional Razumikhin approachThus we provide a new method to investigate the stabilityanalysis for neutral stochastic delayed differential equationswith Markovian switchings In the future we can developthis approach to study the stability for impulsive neutralstochastic delayed differential equations

Remark 9 If system (2) has no Markovian switching and120583(119905) equiv 120583 lt 0 then 119890120593120583 = the condition of Theorem 7 119902 gt(11988821198881)119890120593120583(1 minus 120581)minus119901 becomes 119902 gt (11988821198881)(1 minus 120581)minus119901 which is thesame as the conditions in [46 47] If the model we consideredhas no Markovian switching stochastic disturbance andneutral item then we can obtain the same result as Theorem1 in [60] Thus our result is the generalization of Theorem 1in [60] and the main results in [46 47]

Remark 10 In Theorem 31 of [53] the condition (33)ensured that the coefficient of the estimated upper boundfor the diffusion operator of Lyapunov function is a negative

value and the condition of Theorem 31 in [48] requiredthat the coefficient function of the estimated upper boundfor the diffusion operator of Lyapunov function is a negativedefinite function which are very conservative in the practiceif the system is time-varying system But our conditions allowthis function to take values on R and even allow it to beunbounded Thus our result can be used more widely and isless conservative than the existing results See Example 1 formore details

4 An Example

In this section we will give an example to illustrate thevalidity and significance of our result

Example 1 Now we will consider the following neutral-type stochastic delay differential system with Markovianswitching

119889 [119909 (119905) minus 02119909 (119905 minus 05)]= 119891 (119905 119909 (119905) 119909 (t minus 05) 119903 (119905)) 119889119905

+ 119892 (119905 119909 (119905) 119909 (119905 minus 05) 119903 (119905)) 119889119861119905(24)

where119891 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

=

( 116119905 cos 1199052 minus

32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905) 119894 = 1(18119905 cos 1199052 minus 1) 119909 (119905 minus 05) 119894 = 2

(25)

and119892 (119905 119909 (119905) 119909 (119905 minus 05) 119894)

= (tanh 119905) 119909 (119905 minus 05) 119894 = 1119890minus1199051199092(119905minus05)119909 (119905 minus 05) 119894 = 2

(26)

119903(119905) is a right-continuousMarkov chain on the state spaceS =1 2 with the generator

119876 = [minus1 12 minus2] (27)

Theorem31 in [48] is void in determining the exponentialstability for such a system In fact taking 119881(119905 119909 119894) = |119909|2 and1198881 = 1198882 = 1 then by Itorsquos formula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le 2E(119909 (119905) minus 02119909 (119905 minus 05))sdot [( 1

16119905 cos 1199052 minus32)119909 (119905 minus 05) + 1

16119905 cos 1199052119909 (119905)]+ E |119909 (119905 minus 05)|2 le ( 7

40119905 cos 1199052 minus34)E |119909 (119905)|2

+ ( 340119905 cos 1199052 minus

120)E |119909 (119905 minus 05)|2

(28)

6 Mathematical Problems in Engineering

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le 2E [(119909 (119905) minus 02119909 (119905 minus 05)) (18119905 cos 1199052 minus 1)

sdot 119909 (119905 minus 05)] + E |119909 (119905 minus 05)|2 le (18119905 cos 1199052 minus 1)

sdot E |119909 (119905)|2 + ( 110119905 cos 1199052 minus

25)E |119909 (119905 minus 05)|2

(29)

Obviously the conditions in Theorem 31 [48] do not holdBut we can judge the exponential stability for such system byusing our result Take 119881(119905 119909 119894) = |119909|2 and 1198881 = 1198882 = 1 By Itorsquosformula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le (18119905 cos 1199052 minus 3)E |119909 (119905) 119909 (119905 minus 05)|

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)| + E |119909 (119905 minus 05)|2

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)|

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18 (119902 + 1

2 ) 119905 cos 1199052E |119909 (119905)|2

= [18119905 cos 1199052 (119902 + 1) minus 3 (119902 + 1)2 + 119902]E |119909 (119905)|2

(30)

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le (18119905 cos 1199052 minus 1)E |119909 (119905)|2

+ (18119905 cos 1199052 minus 1)E |119909 (119905 minus 05)|2

+ E |119909 (119905 minus 05)|2

le [(14119905 cos 1199052 minus 2) 119902 + 12 + 119902]E |119909 (119905)|2

= (18119905 cos 1199052 (119902 + 1) + 119902)E |119909 (119905)|2

(31)

Choosing 119902 = 3 then EL119881(119905 119909(119905) 119909(119905 minus 05) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) where 120583(119905) = (12)119905 cos 1199052 minus 1 Here

Ｒ1

Ｒ2

minus015

minus01

minus005

0

005

01

015

10 20 30 40 500t

Figure 1 The state response of Example 1

120583(119905) is an unbounded function taking values in R so wecannot use any former results to judge the mean-squareexponential stability for such system But from our resultwe can determine its stability In fact taking 119879 = 1 thenint119905+1119905 120583(119904)119889119904 = minus12 lt 0 holds for all 119905 ge 0 Thus 119879 isinΩ120583 Additionally 119890120593120583 le 11989005 and 119902minus1119890120593120583(1 minus 119896)minus119901 le 3minus1 times11989005 times 08minus2 lt 1 By Theorem 7 the system is mean squareexponentially stable See Figure 1

5 Conclusion

In this paper we investigate the stability criterion for neu-tral stochastic delay differential equations with Markovianswitching By using stochastic analysis technique and Razu-mikhin approach we overcome the difficulty caused by theneutral item for the related system Finally we derive a newand novel Razumikhin exponential stability criterion Thefeature of the result is that the estimated upper bound forthe diffusion operator of Lyapunov function is allowed totake values on R even if it is allowed to be unboundedThe criterion can reduce some restrictiveness of the relatedresults that are existing in the previous literature An exampleis provided to show the superiority of the new exponentialstability criterion

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This work was supported by the China Postdoctoral ScienceFoundation (2018M632325)

Mathematical Problems in Engineering 7

References

[1] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 998ndash10182017

[2] Y Guo ldquoMean square exponential stability of stochastic delaycellular neural networksrdquo Electronic Journal of QualitativeTheory of Differential Equations no 34 pp 1ndash10 2013

[3] Q Zhu andHWang ldquoOutput feedback stabilization of stochas-tic feedforward systems with unknown control coefficients andunknown output functionrdquo Automatica vol 87 pp 166ndash1752018

[4] H Wang and Q Zhu ldquoFinite-time stabilization of high-orderstochastic nonlinear systems in strict-feedback formrdquoAutomat-ica vol 54 pp 284ndash291 2015

[5] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[6] Y Bai and X Mu ldquoGlobal asymptotic stability of a generalizedSIRS epidemic model with transfer from infectious to suscepti-blerdquo Journal of Applied Analysis and Computation vol 8 no 2pp 402ndash412 2018

[7] Q Zhu S Song andT Tang ldquoMean square exponential stabilityof stochastic nonlinear delay systemsrdquo International Journal ofControl vol 90 no 11 pp 2384ndash2393 2017

[8] Y Guo ldquoGlobal asymptotic stability analysis for integro-differential systems modeling neural networks with delaysrdquoZeitschrift fur Angewandte Mathematik und Physik vol 61 no6 pp 971ndash978 2010

[9] L Gao DWang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[10] Y Guo ldquoExponential stability analysis of travelling waves solu-tions for nonlinear delayed cellular neural networksrdquo Dynami-cal Systems vol 32 no 4 pp 490ndash503 2017

[11] Y Li Y Sun and FMeng ldquoNew criteria for exponential stabilityof switched time-varying systems with delays and nonlineardisturbancesrdquo Nonlinear Analysis Hybrid Systems vol 26 pp284ndash291 2017

[12] B Wang and Q Zhu ldquoStability analysis of Markov switchedstochastic differential equations with both stable and unstablesubsystemsrdquo Systems amp Control Letters vol 105 pp 55ndash61 2017

[13] L GWang K P Xu and QW Liu ldquoOn the stability of a mixedfunctional equation deriving fromadditive quadratic and cubicmappingsrdquo Acta Mathematica Sinica vol 30 no 6 pp 1033ndash1049 2014

[14] Q Zhu S Song and P Shi ldquoEffect of noise on the solutions ofnon-linear delay systemsrdquo IET Control Theory amp Applicationsvol 12 no 13 pp 1822ndash1829 2018

[15] G Liu S Xu Y Wei Z Qi and Z Zhang ldquoNew insightinto reachable set estimation for uncertain singular time-delaysystemsrdquo Applied Mathematics and Computation vol 320 pp769ndash780 2018

[16] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[17] Q Zhu ldquoAsymptotic stability in the pth moment for stochasticdifferential equations with Levy noiserdquo Journal of MathematicalAnalysis and Applications vol 416 no 1 pp 126ndash142 2014

[18] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[19] J Hu and A Xu ldquoOn stability of F-Gorenstein flat categoriesrdquoAlgebra Colloquium vol 23 no 2 pp 251ndash262 2016

[20] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[21] H Wang and Q Zhu ldquoGlobal Stabilization of StochasticNonlinear Systems via C 1 and C infin Controllersrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 11 pp 5880ndash5887 2017

[22] Y Guo ldquoMean square global asymptotic stability of stochasticrecurrent neural networks with distributed delaysrdquo AppliedMathematics andComputation vol 215 no 2 pp 791ndash795 2009

[23] Q Zhu ldquoStability analysis of stochastic delay differential equa-tions with Levy noiserdquo Systems amp Control Letters vol 118 pp62ndash68 2018

[24] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[25] W Qi Y Kao X Gao and Y Wei ldquoController design fortime-delay system with stochastic disturbance and actuatorsaturation via a new criterionrdquo Applied Mathematics and Com-putation vol 320 pp 535ndash546 2018

[26] L G Wang and B Liu ldquoFuzzy stability of a functional equationderiving from additive quadratic cubic and quartic functionsrdquoActa Mathematica Sinica vol 55 no 5 pp 841ndash854 2012

[27] Q Zhu and B Song ldquoExponential stability of impulsivenonlinear stochastic differential equations with mixed delaysrdquoNonlinear Analysis Real World Applications vol 12 no 5 pp2851ndash2860 2011

[28] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[29] Y-H Feng and C-M Liu ldquoStability of steady-state solutionsto Navier-Stokes-Poisson systemsrdquo Journal of MathematicalAnalysis and Applications vol 462 no 2 pp 1679ndash1694 2018

[30] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[31] T Li and Y F Yang ldquoStability of large steady-state solutionsto non-isentropic Euler-Maxwell systemsrdquo Journal of NanjingNormal University vol 40 no 4 pp 26ndash35 2017

[32] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquo Ukrainian Mathematical Journal vol 69 no 8 pp1049ndash1060 2017

[33] Q Zhu J Cao andR Rakkiyappan ldquoExponential input-to-statestability of stochastic Cohen-Grossberg neural networks withmixed delaysrdquoNonlinearDynamics vol 79 no 2 pp 1085ndash10982015

[34] L GWang andB Liu ldquoTheHyers-Ulam stability of a functionalequation deriving from quadratic and cubic functions in quasi-120573-normed spacesrdquo Acta Mathematica Sinica vol 26 no 12 pp2335ndash2348 2010

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le 2E [(119909 (119905) minus 02119909 (119905 minus 05)) (18119905 cos 1199052 minus 1)

sdot 119909 (119905 minus 05)] + E |119909 (119905 minus 05)|2 le (18119905 cos 1199052 minus 1)

sdot E |119909 (119905)|2 + ( 110119905 cos 1199052 minus

25)E |119909 (119905 minus 05)|2

(29)

Obviously the conditions in Theorem 31 [48] do not holdBut we can judge the exponential stability for such system byusing our result Take 119881(119905 119909 119894) = |119909|2 and 1198881 = 1198882 = 1 By Itorsquosformula

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 1)le (18119905 cos 1199052 minus 3)E |119909 (119905) 119909 (119905 minus 05)|

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)| + E |119909 (119905 minus 05)|2

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18119905 cos 1199052E |119909 (119905) 119909 (119905)|

le [(18119905 cos 1199052 minus 3) 119902 + 12 + 119902]E |119909 (119905)|2

+ 18 (119902 + 1

2 ) 119905 cos 1199052E |119909 (119905)|2

= [18119905 cos 1199052 (119902 + 1) minus 3 (119902 + 1)2 + 119902]E |119909 (119905)|2

(30)

and

EL119881(119905 119909 (119905) 119909 (119905 minus 05) 2)le (18119905 cos 1199052 minus 1)E |119909 (119905)|2

+ (18119905 cos 1199052 minus 1)E |119909 (119905 minus 05)|2

+ E |119909 (119905 minus 05)|2

le [(14119905 cos 1199052 minus 2) 119902 + 12 + 119902]E |119909 (119905)|2

= (18119905 cos 1199052 (119902 + 1) + 119902)E |119909 (119905)|2

(31)

Choosing 119902 = 3 then EL119881(119905 119909(119905) 119909(119905 minus 05) 119903(119905)) le120583(119905)E119881(119905 119909(119905) 119903(119905)) where 120583(119905) = (12)119905 cos 1199052 minus 1 Here

Ｒ1

Ｒ2

minus015

minus01

minus005

0

005

01

015

10 20 30 40 500t

Figure 1 The state response of Example 1

120583(119905) is an unbounded function taking values in R so wecannot use any former results to judge the mean-squareexponential stability for such system But from our resultwe can determine its stability In fact taking 119879 = 1 thenint119905+1119905 120583(119904)119889119904 = minus12 lt 0 holds for all 119905 ge 0 Thus 119879 isinΩ120583 Additionally 119890120593120583 le 11989005 and 119902minus1119890120593120583(1 minus 119896)minus119901 le 3minus1 times11989005 times 08minus2 lt 1 By Theorem 7 the system is mean squareexponentially stable See Figure 1

5 Conclusion

In this paper we investigate the stability criterion for neu-tral stochastic delay differential equations with Markovianswitching By using stochastic analysis technique and Razu-mikhin approach we overcome the difficulty caused by theneutral item for the related system Finally we derive a newand novel Razumikhin exponential stability criterion Thefeature of the result is that the estimated upper bound forthe diffusion operator of Lyapunov function is allowed totake values on R even if it is allowed to be unboundedThe criterion can reduce some restrictiveness of the relatedresults that are existing in the previous literature An exampleis provided to show the superiority of the new exponentialstability criterion

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This work was supported by the China Postdoctoral ScienceFoundation (2018M632325)

Mathematical Problems in Engineering 7

References

[1] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 998ndash10182017

[2] Y Guo ldquoMean square exponential stability of stochastic delaycellular neural networksrdquo Electronic Journal of QualitativeTheory of Differential Equations no 34 pp 1ndash10 2013

[3] Q Zhu andHWang ldquoOutput feedback stabilization of stochas-tic feedforward systems with unknown control coefficients andunknown output functionrdquo Automatica vol 87 pp 166ndash1752018

[4] H Wang and Q Zhu ldquoFinite-time stabilization of high-orderstochastic nonlinear systems in strict-feedback formrdquoAutomat-ica vol 54 pp 284ndash291 2015

[5] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[6] Y Bai and X Mu ldquoGlobal asymptotic stability of a generalizedSIRS epidemic model with transfer from infectious to suscepti-blerdquo Journal of Applied Analysis and Computation vol 8 no 2pp 402ndash412 2018

[7] Q Zhu S Song andT Tang ldquoMean square exponential stabilityof stochastic nonlinear delay systemsrdquo International Journal ofControl vol 90 no 11 pp 2384ndash2393 2017

[8] Y Guo ldquoGlobal asymptotic stability analysis for integro-differential systems modeling neural networks with delaysrdquoZeitschrift fur Angewandte Mathematik und Physik vol 61 no6 pp 971ndash978 2010

[9] L Gao DWang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[10] Y Guo ldquoExponential stability analysis of travelling waves solu-tions for nonlinear delayed cellular neural networksrdquo Dynami-cal Systems vol 32 no 4 pp 490ndash503 2017

[11] Y Li Y Sun and FMeng ldquoNew criteria for exponential stabilityof switched time-varying systems with delays and nonlineardisturbancesrdquo Nonlinear Analysis Hybrid Systems vol 26 pp284ndash291 2017

[12] B Wang and Q Zhu ldquoStability analysis of Markov switchedstochastic differential equations with both stable and unstablesubsystemsrdquo Systems amp Control Letters vol 105 pp 55ndash61 2017

[13] L GWang K P Xu and QW Liu ldquoOn the stability of a mixedfunctional equation deriving fromadditive quadratic and cubicmappingsrdquo Acta Mathematica Sinica vol 30 no 6 pp 1033ndash1049 2014

[14] Q Zhu S Song and P Shi ldquoEffect of noise on the solutions ofnon-linear delay systemsrdquo IET Control Theory amp Applicationsvol 12 no 13 pp 1822ndash1829 2018

[15] G Liu S Xu Y Wei Z Qi and Z Zhang ldquoNew insightinto reachable set estimation for uncertain singular time-delaysystemsrdquo Applied Mathematics and Computation vol 320 pp769ndash780 2018

[16] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[17] Q Zhu ldquoAsymptotic stability in the pth moment for stochasticdifferential equations with Levy noiserdquo Journal of MathematicalAnalysis and Applications vol 416 no 1 pp 126ndash142 2014

[18] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[19] J Hu and A Xu ldquoOn stability of F-Gorenstein flat categoriesrdquoAlgebra Colloquium vol 23 no 2 pp 251ndash262 2016

[20] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[21] H Wang and Q Zhu ldquoGlobal Stabilization of StochasticNonlinear Systems via C 1 and C infin Controllersrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 11 pp 5880ndash5887 2017

[22] Y Guo ldquoMean square global asymptotic stability of stochasticrecurrent neural networks with distributed delaysrdquo AppliedMathematics andComputation vol 215 no 2 pp 791ndash795 2009

[23] Q Zhu ldquoStability analysis of stochastic delay differential equa-tions with Levy noiserdquo Systems amp Control Letters vol 118 pp62ndash68 2018

[24] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[25] W Qi Y Kao X Gao and Y Wei ldquoController design fortime-delay system with stochastic disturbance and actuatorsaturation via a new criterionrdquo Applied Mathematics and Com-putation vol 320 pp 535ndash546 2018

[26] L G Wang and B Liu ldquoFuzzy stability of a functional equationderiving from additive quadratic cubic and quartic functionsrdquoActa Mathematica Sinica vol 55 no 5 pp 841ndash854 2012

[27] Q Zhu and B Song ldquoExponential stability of impulsivenonlinear stochastic differential equations with mixed delaysrdquoNonlinear Analysis Real World Applications vol 12 no 5 pp2851ndash2860 2011

[28] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[29] Y-H Feng and C-M Liu ldquoStability of steady-state solutionsto Navier-Stokes-Poisson systemsrdquo Journal of MathematicalAnalysis and Applications vol 462 no 2 pp 1679ndash1694 2018

[30] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[31] T Li and Y F Yang ldquoStability of large steady-state solutionsto non-isentropic Euler-Maxwell systemsrdquo Journal of NanjingNormal University vol 40 no 4 pp 26ndash35 2017

[32] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquo Ukrainian Mathematical Journal vol 69 no 8 pp1049ndash1060 2017

[33] Q Zhu J Cao andR Rakkiyappan ldquoExponential input-to-statestability of stochastic Cohen-Grossberg neural networks withmixed delaysrdquoNonlinearDynamics vol 79 no 2 pp 1085ndash10982015

[34] L GWang andB Liu ldquoTheHyers-Ulam stability of a functionalequation deriving from quadratic and cubic functions in quasi-120573-normed spacesrdquo Acta Mathematica Sinica vol 26 no 12 pp2335ndash2348 2010

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

References

[1] X Zheng Y Shang and X Peng ldquoOrbital stability of periodictraveling wave solutions to the generalized Zakharov equa-tionsrdquo Acta Mathematica Scientia vol 37 no 4 pp 998ndash10182017

[2] Y Guo ldquoMean square exponential stability of stochastic delaycellular neural networksrdquo Electronic Journal of QualitativeTheory of Differential Equations no 34 pp 1ndash10 2013

[3] Q Zhu andHWang ldquoOutput feedback stabilization of stochas-tic feedforward systems with unknown control coefficients andunknown output functionrdquo Automatica vol 87 pp 166ndash1752018

[4] H Wang and Q Zhu ldquoFinite-time stabilization of high-orderstochastic nonlinear systems in strict-feedback formrdquoAutomat-ica vol 54 pp 284ndash291 2015

[5] G Wang ldquoExistence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spacesrdquo Jour-nal of Inequalities and Applications vol 239 pp 1ndash14 2015

[6] Y Bai and X Mu ldquoGlobal asymptotic stability of a generalizedSIRS epidemic model with transfer from infectious to suscepti-blerdquo Journal of Applied Analysis and Computation vol 8 no 2pp 402ndash412 2018

[7] Q Zhu S Song andT Tang ldquoMean square exponential stabilityof stochastic nonlinear delay systemsrdquo International Journal ofControl vol 90 no 11 pp 2384ndash2393 2017

[8] Y Guo ldquoGlobal asymptotic stability analysis for integro-differential systems modeling neural networks with delaysrdquoZeitschrift fur Angewandte Mathematik und Physik vol 61 no6 pp 971ndash978 2010

[9] L Gao DWang and G Wang ldquoFurther results on exponentialstability for impulsive switched nonlinear time-delay systemswith delayed impulse effectsrdquo Applied Mathematics and Com-putation vol 268 pp 186ndash200 2015

[10] Y Guo ldquoExponential stability analysis of travelling waves solu-tions for nonlinear delayed cellular neural networksrdquo Dynami-cal Systems vol 32 no 4 pp 490ndash503 2017

[11] Y Li Y Sun and FMeng ldquoNew criteria for exponential stabilityof switched time-varying systems with delays and nonlineardisturbancesrdquo Nonlinear Analysis Hybrid Systems vol 26 pp284ndash291 2017

[12] B Wang and Q Zhu ldquoStability analysis of Markov switchedstochastic differential equations with both stable and unstablesubsystemsrdquo Systems amp Control Letters vol 105 pp 55ndash61 2017

[13] L GWang K P Xu and QW Liu ldquoOn the stability of a mixedfunctional equation deriving fromadditive quadratic and cubicmappingsrdquo Acta Mathematica Sinica vol 30 no 6 pp 1033ndash1049 2014

[14] Q Zhu S Song and P Shi ldquoEffect of noise on the solutions ofnon-linear delay systemsrdquo IET Control Theory amp Applicationsvol 12 no 13 pp 1822ndash1829 2018

[15] G Liu S Xu Y Wei Z Qi and Z Zhang ldquoNew insightinto reachable set estimation for uncertain singular time-delaysystemsrdquo Applied Mathematics and Computation vol 320 pp769ndash780 2018

[16] W Sun Y Wang and R Yang ldquoL2 disturbance attenuation fora class of time delay Hamiltonian systemsrdquo Journal of SystemsScience amp Complexity vol 24 no 4 pp 672ndash682 2011

[17] Q Zhu ldquoAsymptotic stability in the pth moment for stochasticdifferential equations with Levy noiserdquo Journal of MathematicalAnalysis and Applications vol 416 no 1 pp 126ndash142 2014

[18] Y Guo ldquoGlobal stability analysis for a class of Cohen-Grossbergneural network modelsrdquo Bulletin of the Korean MathematicalSociety vol 49 no 6 pp 1193ndash1198 2012

[19] J Hu and A Xu ldquoOn stability of F-Gorenstein flat categoriesrdquoAlgebra Colloquium vol 23 no 2 pp 251ndash262 2016

[20] W W Sun ldquoStabilization analysis of time-delay Hamiltoniansystems in the presence of saturationrdquoAppliedMathematics andComputation vol 217 no 23 pp 9625ndash9634 2011

[21] H Wang and Q Zhu ldquoGlobal Stabilization of StochasticNonlinear Systems via C 1 and C infin Controllersrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 62 no 11 pp 5880ndash5887 2017

[22] Y Guo ldquoMean square global asymptotic stability of stochasticrecurrent neural networks with distributed delaysrdquo AppliedMathematics andComputation vol 215 no 2 pp 791ndash795 2009

[23] Q Zhu ldquoStability analysis of stochastic delay differential equa-tions with Levy noiserdquo Systems amp Control Letters vol 118 pp62ndash68 2018

[24] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear AnalystsNonlinear Analysis Modelling and Control vol 19 no 4 pp626ndash645 2014

[25] W Qi Y Kao X Gao and Y Wei ldquoController design fortime-delay system with stochastic disturbance and actuatorsaturation via a new criterionrdquo Applied Mathematics and Com-putation vol 320 pp 535ndash546 2018

[26] L G Wang and B Liu ldquoFuzzy stability of a functional equationderiving from additive quadratic cubic and quartic functionsrdquoActa Mathematica Sinica vol 55 no 5 pp 841ndash854 2012

[27] Q Zhu and B Song ldquoExponential stability of impulsivenonlinear stochastic differential equations with mixed delaysrdquoNonlinear Analysis Real World Applications vol 12 no 5 pp2851ndash2860 2011

[28] F Li andYBao ldquoUniformstability of the solution for amemory-type elasticity system with nonhomogeneous boundary controlconditionrdquo Journal of Dynamical and Control Systems vol 23no 2 pp 301ndash315 2017

[29] Y-H Feng and C-M Liu ldquoStability of steady-state solutionsto Navier-Stokes-Poisson systemsrdquo Journal of MathematicalAnalysis and Applications vol 462 no 2 pp 1679ndash1694 2018

[30] X Zheng Y Shang and X Peng ldquoOrbital stability of soli-tary waves of the coupled Klein-Gordon-Zakharov equationsrdquoMathematical Methods in the Applied Sciences vol 40 no 7 pp2623ndash2633 2017

[31] T Li and Y F Yang ldquoStability of large steady-state solutionsto non-isentropic Euler-Maxwell systemsrdquo Journal of NanjingNormal University vol 40 no 4 pp 26ndash35 2017

[32] Y Guo ldquoGlobally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse and time-varyingdelaysrdquo Ukrainian Mathematical Journal vol 69 no 8 pp1049ndash1060 2017

[33] Q Zhu J Cao andR Rakkiyappan ldquoExponential input-to-statestability of stochastic Cohen-Grossberg neural networks withmixed delaysrdquoNonlinearDynamics vol 79 no 2 pp 1085ndash10982015

[34] L GWang andB Liu ldquoTheHyers-Ulam stability of a functionalequation deriving from quadratic and cubic functions in quasi-120573-normed spacesrdquo Acta Mathematica Sinica vol 26 no 12 pp2335ndash2348 2010

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

[35] M Li and J Wang ldquoExploring delayed Mittag-Leffler typematrix functions to study finite time stability of fractional delaydifferential equationsrdquo Applied Mathematics and Computationvol 324 pp 254ndash265 2018

[36] Y Guo ldquoNontrivial periodic solutions of nonlinear functionaldifferential systems with feedback controlrdquo Turkish Journal ofMathematics vol 34 no 1 pp 35ndash44 2010

[37] L Li F Meng and P Ju ldquoSome new integral inequalities andtheir applications in studying the stability of nonlinear integro-differential equations with time delayrdquo Journal of MathematicalAnalysis and Applications vol 377 no 2 pp 853ndash862 2011

[38] MMariton Jump Linear Systems in Automatica Control MarcelDekker 1990

[39] R Song andQ Zhu ldquoStability of linear stochastic delay differen-tial equationswith infiniteMarkovian switchingsrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 3 pp 825ndash837 2018

[40] B Wang and Q Zhu ldquoStability analysis of semi-Markovswitched stochastic systemsrdquo Automatica vol 94 pp 72ndash802018

[41] Q X Zhu and Q Y Zhang ldquoPth moment exponential stabil-isation of hybrid stochastic differential equations by feedbackcontrols based on discrete-time state observations with a timedelayrdquo IET Control Theory amp Applications vol 11 no 12 pp1992ndash2003 2017

[42] Q Zhu R Rakkiyappan and A Chandrasekar ldquoStochasticstability ofMarkovian jump BAMneural networks with leakagedelays and impulse controlrdquo Neurocomputing vol 136 pp 136ndash151 2014

[43] Q Zhu and J Cao ldquoStability of Markovian jump neural net-works with impulse control and time varying delaysrdquoNonlinearAnalysis Real World Applications vol 13 no 5 pp 2259ndash22702012

[44] Y Liu J H Park B Guo F Fang and F Zhou ldquoEvent-triggered dissipative synchronization for Markovian jump neu-ral networks with general transition probabilitiesrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 13 pp3893ndash3908 2018

[45] R Zhang D Zeng J H Park Y Liu and S Zhong ldquoA newapproach to stochastic stability of markovian neural networkswith generalized transition ratesrdquo IEEE Transactions on NeuralNetworks and Learning Systems pp 1ndash12 2018

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

[47] X Mao ldquoRazumikhin-type theorems on exponential stabilityof neutral stochastic functional-differential equationsrdquo SIAMJournal on Mathematical Analysis vol 28 no 2 pp 389ndash4011997

[48] H Chen P Shi and C-C Lim ldquoStability analysis for neutralstochastic delay systems with Markovian switchingrdquo Systems ampControl Letters vol 110 pp 38ndash48 2017

[49] Q Zhu and J Cao ldquo pth moment exponential synchronizationfor stochastic delayed Cohen-Grossberg neural networks withMarkovian switchingrdquo Nonlinear Dynamics vol 67 no 1 pp829ndash845 2012

[50] Y Chen W X Zheng and A Xue ldquoA new result on stabilityanalysis for stochastic neutral systemsrdquo Automatica vol 46 no12 pp 2100ndash2104 2010

[51] L Shaikhet ldquoSome new aspects of Lyapunov-type theorems forstochastic differential equations of neutral typerdquo SIAM Journalon Control andOptimization vol 48 no 7 pp 4481ndash4499 2010

[52] Q Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012

[53] Y Xu and Z He ldquoExponential stability of neutral stochasticdelay differential equations with Markovian switchingrdquoAppliedMathematics Letters vol 52 pp 64ndash73 2016

[54] K Liu and X Xia ldquoOn the exponential stability in mean squareof neutral stochastic functional differential equationsrdquo Systemsamp Control Letters vol 37 no 4 pp 207ndash215 1999

[55] Q Zhu ldquoRazumikhin-type theorem for stochastic functionaldifferential equations with Levy noise and Markov switchingrdquoInternational Journal of Control vol 90 no 8 pp 1703ndash17122017

[56] Q Zhu ldquopthmoment exponential stability of impulsive stochas-tic functional differential equations with Markovian switchingrdquoJournal of The Franklin Institute vol 351 no 7 pp 3965ndash39862014

[57] J Bao Z Hou and C Yuan ldquoStability in distribution ofneutral stochastic differential delay equations with Markovianswitchingrdquo Statistics amp Probability Letters vol 79 no 15 pp1663ndash1673 2009

[58] X Li and X Mao ldquoA note on almost sure asymptotic stability ofneutral stochastic delay differential equations with Markovianswitchingrdquo Automatica vol 48 no 9 pp 2329ndash2334 2012

[59] X Mao Y Shen and C Yuan ldquoAlmost surely asymptoticstability of neutral stochastic differential delay equations withMarkovian switchingrdquo Stochastic Processes and Their Applica-tions vol 118 no 8 pp 1385ndash1406 2008

[60] B Zhou andA V Egorov ldquoRazumikhin and Krasovskii stabilitytheorems for time-varying time-delay systemsrdquo Automaticavol 71 pp 281ndash291 2016

[61] B Zhou and W Luo ldquoImproved Razumikhin and Krasovskiistability criteria for time-varying stochastic time-delay sys-temsrdquo Automatica vol 89 pp 382ndash391 2018

[62] F Mazenc M Malisoff and S-I Niculescu ldquoStability anal-ysis for systems with time-varying delay Trajectory basedapproachrdquo in Proceedings of the 54th IEEE Conference onDecision and Control (CDC rsquo15) pp 1811ndash1816 Osaka JapanDecember 2015

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MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom