Research Article Mean Square Stability of Impulsive Stochastic Differential Systems

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1 International Differential Equations Volume 011, Article ID , 13 pages doi: /011/ Research Article Mean Square Stability of Impulsive Stochastic Differential Systems Shujie Yang, Bao Shi, and Mo Li Institute of System Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, Shandong 64001, China Correspondence should be addressed to Shujie Yang, Received 4 December 010; Revised 17 May 011; Accepted 6 May 011 Academic Editor: Xingfu Zou Copyright q 011 Shujie Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Based on Lyapunov-Krasovsii functional method and stochastic analysis theory, we obtain some new delay-dependent criteria ensuring mean square stability of a class of impulsive stochastic equations. Numerical examples are given to illustrate the effectiveness of the theoretical results. 1. Introduction It is recognized that the theory of impulsive systems provides a natural framewor for the mathematical modeling of many real world phenomena, and impulsive dynamical systems have attracted considerable interest in science and engineering during the past decades. Two classical monographs are Lashmiantham et al. 1 and Bainov and Simeonov. In general, an impulsive dynamical system can be viewed as a hybrid one comprised of three components: a continuous-time differential equation, which governs the motion of the dynamical systems between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs and a criterion for determining when the states of the systems are to be reset, see Chen and Zheng 3. Stability properties of impulsive systems have been extensively studied in the literatures. We refer to Li et al. 4, 5, Lietal. 6, Yang 7, Autonio, and Alfonso 8 and the references therein. Besides impulsive effects, a practical system is usually affected by external stochastic perturbations. Stochastic perturbation is also a factor that maes systems unstable. Recently, stochastic modeling has come to play an important role in many branches of science and industry. An area of particular interest has been stability analysis of impulsive systems with

2 International Differential Equations stochastic perturbation. In Yang et al. 9 and Chen et al. 10, the stability properties of nonlinear impulsive stochastic systems are studied using Lyapunov function methods. In Mao et al. 11, a linear matrix inequality approach is proposed for stability analysis of linear uncertain impulsive stochastic systems. However, to the best of our nowledge, there are only few results about this problem. This paper is inspired by Yang et al. 9, in which the authors considered the problems of stability or robust stabilization for impulsive time delay systems. Unfortunately, they need all the impulsive time sequences to satisfy some strict conditions, that is, the length of the intervals between two jumping time instants must have upper bound or lower bound. But in practical systems, it is always impossible or difficult to obtain it. In this article, by using Lyapunov function methods, together with stochastic analysis, we focus on the mean square stability of trival solution of a class of nonlinear impulsive stochastic time-delay differential systems. We obtain some new conditions ensuring mean square stability of trival solution of the impulsive stochastic differential systems with time-delay. This paper improved some related results.. Preliminaries Throughout this paper, unless explicitly given, for symmetric matrices A and B, the notion A B A >B, A B, A<B means A B is positive semidefinite positive definite, negative semidefinite, negative definite matrix. λ max λ min represents the maximum minimum eigenvalue of the corresponding matrix, respectively. denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let Ω, F, F t } t 0,P be a complete probability space with a filtration F t } t 0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC τ, 0, R n denote the set of piecewise right continuous function φ : τ, 0 R n with the norm defined by φ τ sup τ θ 0 φ θ, where τ is a nown positive constant, PC δ ϕ ϕ PC τ, 0, R n, ϕ τ δ}, PC b F 0 τ, 0, R n denote the family of all F 0 -measurable PC τ, 0, R n -valued stochastic process ϕ ϕ s : τ s 0} with sup τ s 0 E ϕ s } <, where E } represents the mathematical expectation operator with respect to the probability measure P, PC b F 0 δ ϕ ϕ PC b F 0 τ, 0, R n, sup τ s 0 E ϕ s } δ, L denote the well-nown L-operator given by the Itô s formula. In this paper, we consider a class of Itô impulsive stochastic differential systems with time delay dx t f t, x t,x t dt g t, x t,x t dω t, t t 0,t/ t, ( ( x t H x t )), 1,,..., x t 0 θ ϕ θ, θ τ, 0,.1 where the initial value ϕ PC b F 0 δ, the fixed impulsive time moments t satisfy 0 t 0 < t 1 <t < <t < t as. x t R n is the system state, f C R R n R n, R n, g C R R n R n, R n m. ω t R m is an standard Brownian motion defined on

3 International Differential Equations 3 the complete probability space Ω, F, F t } t 0,P. Besides, we assume that H 0 0, 1,,..., f t, 0, 0 0, g t, 0, 0 0and ( ( )) ( H x t γ x t, ) γ 0, 1,,.... In the following, we will divide three cases to consider the mean square stability of system.1. We denote by N inf β and N sup β the class of impulsive time sequences that satisfy inf t t 1 } β and sup t t 1 } β, respectively. We need the following lemma and definitions. Lemma.1 Chaplygin Comparison Theorem, see Shi et al. 1. Assume that f, F C G, g R and f t, x <F t, x, t, x G..3 If φ t (t U 1 ) and Φ t (t U ) are the solutions of Cauchy problems ẋ f t, x, t, x G, x τ ξ, ẋ F t, x, t, x G,.4 x τ ξ, respectively, then for t τ, U 1 U, φ t < Φ t.5 and for t,τ U 1 U, φ t > Φ t..6 Definition.. For a given class N of admissible impulsive time sequence, the solution of.1 is called mean squarely stable if for any ε>0, there exists a scalar δ>0, such that the initial function ϕ PC b F 0 δ implies E x t } <ε, t t 0 for all admissible time sequence in N. Definition.3 see Yang et al. 9. The function V : t 0 τ, R n V 1, if R belongs to class 1 the function V t, x is continuously differentiable in t and twice continuously differentiable in x on each of the sets t 1,t R n, 1,,... and for all t t 0, V t, 0 0, V t, x is locally Lipschitaian in x,

4 4 International Differential Equations 3 for each 1,,..., there exist finite limits lim t,y t,x V ( t, y ) V ( t,x), V ( t,x) V t,x. lim ( t,y t,x V t, y ) V ( t,x),.7 3. Main Results Theorem 3.1. Assume that there exist scalars λ >λ 1 > 0, λ τ > 0, β>0, λ 0, ρ>0 matrix P>0 and Lyapunov-Krasovsii functional V t, x t V 1,, such that C1 λ 1 x t V t, x t λ x t τ, C ELV t, x t λev t, x t λ τ EV t, x t,t t 1,t, 1,,..., whenever EV t, x t μ ρ EV t, x t, C3 μ sup N λ λ /λ 1 γ } > 1 and λ μ ρ λ τ ln μ ρ /β, then the trivial solution of system.1 is mean squarely stable over N inf τ β. Proof. For any given ε>0, choose 0 <δ λ 1 / μ ρ λ ε. We assume that the initial function ϕ PC b F 0 δ and denote the solution x t, t 0,ϕ of system.1 through t 0,ϕ by x t. Inthe following, we will prove that x t is mean square stable over N inf τ β. For V t, x t V 1,, by Itô formula, for t / t, 1,,..., we have dv t, x t LV t, x t dt V x t, x t g t, x t dω t, 3.1 where LV t, x t V t t, x t V x t, x t f 1/ tr g T V xx g. For t t 1,t, 1,,..., integrate 3.1 from t 1 to t, we have t t V t, x t V t 1,x t 1 LV s, x s ds t 1 V x s, x s g s, x s dω s. t 1 3. Taing the mathematical expectation of both sides of the above equation, we obtain t EV t, x t EV t 1,x t 1 ELV s, x s ds. t So for s t, t Δt with t Δt t 1,t and Δt >0, if EV s, x s μ ρ EV s, x s, then we have by C EV t Δt, x t Δt EV t, x t t Δt t t Δt t ELV s, x s ds λev s, x s λ τ EV s, x s ds. 3.4

5 International Differential Equations 5 In what follows, we first prove that for t t 0 τ, t 1, EV t, x t λ 1 ε. 3.5 Obviously, for t t 0 τ, t 0,by C1 and x t0 PC b F 0 δ,weobtain } EV t, x t λ E x t0 τ λ δ λ 1 μ ρ ε <λ 1 ε. 3.6 Now it needs only to prove that for t t 0,t 1, 3.5 holds. Otherwise, there exists s t 0,t 1, such that EV s, x s >λ 1 ε. 3.7 Set s 1 inf t t 0,t 1 : EV t, x t >λ 1 ε }, 3.8 then by 3.6, 3.7, and the continuity of EV t, x t on t 0,t 1, we now that s 1 t 0,t 1, EV s 1,x s 1 λ 1 ε, 3.9 and for t t 0 τ, s 1, 3.5 holds. Set s sup t t 0,s 1 : EV t, x t λ } 1 μ ρ ε, 3.10 then by 3.6 and the continuity of EV t, x t, we have s t 0,s 1, EV s,x s λ 1 μ ρ ε, 3.11 and for t s,s 1, EV t, x t λ 1 ε ( μ ρ ) EV t, x t, 3.1 which implies with 3.4 and C3 that for t s,s 1, D EV t, x t λev t, x t λ τ EV t, x t ( λ λ τ ( μ ρ )) EV t, x t This is a contradiction with 3.9 and 3.11.

6 6 International Differential Equations Now, we assume that, for t t m1,t m, m 1,,...,, 3.5 holds. For m 1, we will show that 3.5 holds. To this end, we first prove that for θ τ, 0, EV ( t θ, x( t θ)) λ 1 μ ρ ε Noticing t τ, t t 1,t, we assume that there exists some s t τ, t, such that EV ( s,x ( s )) > λ 1 μ ρ ε, 3.15 then there are two cases to be considered. i For all t t 1,s, EV t,x t > λ 1 / μ ρ ε. Hence, for t t 1,s, 3.1 and 3.13 hold, which follows by C3, 3.5, and Lemma.1, EV ( s,x ( s )) exp ( λ ( μ ρ ) λ τ ) s t1 } EV t 1,x t 1 λ 1 ε exp ( λ ( μ ρ ) λ τ ) t t 1 τ } 3.16 λ 1 μ ρ ε, this is a contradiction with the assumption. ii There exists some t t 1,s, such that EV t, x t λ 1 / μ ρ ε.set s 1 sup t t 1,s : EV t, x t λ } 1 μ ρ ε, 3.17 then s 1 t 1,s, EV s 1,x s 1 λ 1 μ ρ ε 3.18 and for t s 1,s, 3.1 and 3.13 hold, which is a contradiction with 3.15 and 3.18, thatis, 3.14 holds. By.1,., and 3.14, we have EV t,x t EV ( t,h ( x ( t ))) λ E λ γ λ 1 H ( x ( t )) } τ λ γ E x ( t sup EV ( t θ, x ( t θ)) λ μ ρ ε <λ 1 ε. τ θ 0 λ 1 ) } τ 3.19

7 International Differential Equations 7 Now we will prove that 3.5 holds for t t,t 1. Otherwise, there exists some t t,t 1, such that 3.7 holds. Let s 1 inf t t,t 1 : EV t, x t >λ 1 ε }. 3.0 Then by 3.14, 3.19, and the continuity of EV t, x t on t,t 1, we now that s 1 t,t 1 and EV s 1,x s 1 λ 1 ε. If there exists t t,s 1, such that EV t, x t λ 1 /μ ε, then let s sup t t,s 1 : EV t, x t λ } 1 μ ρ ε. 3.1 Otherwise, let s t. Then for t s,s 1,weobtain 3.1 and 3.13, which follows a contradiction. By mathematical induction, 3.5 holds for any m 1,,..., which implies that system.1 is mean squarely stable. If substituting condition C 1 λ 1 x t V t, x t λ x t for C1 in Theorem 3.1, then we have the following result. Theorem 3.. Assume that there exist scalars λ >λ 1 > 0, λ τ > 0, λ 0, ρ>0, matrix P>0 and Lyapunov-Krasovsii functional V t, x t V 1,, such that conditions (C 1), (C), and (C3) hold, then the trivial solution of system.1 is mean square stable over N inf β. Proof. The proof is similar to Theorem 3.1, so we omit it. The proof is complete. Remar 3.3. Comparing the results in Theorems 3.1 and 3., we find the influence of the time delay on the mean square stability of system.1. Remar 3.4. When μ>1, the impulses which may be destabilizing, so we require the impulses should not happen so frequently. When μ 1, we have the following results. Theorem 3.5. Assume that there exist scalars λ > λ 1 > 0, λ τ > 0, λ 0, matrixp > 0 and Lyapunov-Krasovsii functional V t, x t V 1,, such that condition (C1) and C ELV t, x t λev t, x t λ τ EV t, x t, t t 1,t, 1,,... whenever EV t, x t EV t, x t, C 3 μ sup N λ λ /λ 1 γ } 1, λ λ τ 0 hold, then the trivial solution of system.1 is mean squarely stable over any impulsive sequences. Proof. For any given ε>0, choose 0 <δ λ 1 /λ ε. We assume that the initial function ϕ PC b F 0 δ. In what follows, we first prove that for t t 0, 3.5 holds. Obviously, for t t 0 τ, t 0,by C1 and x t0 PC b F 0 δ,weobtain } EV t, x t λ E x t0 τ λ δ λ 1 ε. 3.

8 8 International Differential Equations Now we should prove that 3.5 holds. Otherwise, there exists s t 0,t 1, such that 3.7 holds. By 3. and the continuity of EV t, x t on t 0,t 1, we now there exist t t 0,t 1 and small scalar ρ>0, such that ( ( )) EV t, x t λ 1 ε 3.3 and for every t 1, t t, t ρ, t 1 < t, ( )) ( )) EV t 1,x (t 1 < EV t,x (t. 3.4 Let s inft t 0,t 1 : EV t, x t λ 1 ε, EV u, x u > λ 1 ε,u t, t ρ 1 t 0,t 1, EV t 1,x t 1 < EV t,x t, for every t 1, t t, t ρ 1, t 1 < t }, where ρ 1 > 0 is some scalar. Then s, s ρ 1 t 0, t 1 and for t s, s ρ 1, EV t, x t EV t, x t, 3.5 which implies with C and C 3 that for t s, s ρ 1, D EV t, x t λev t, x t λ τ EV t, x t λ λ τ EV t, x t This is a contradiction with the fact EV s ρ 1,x s ρ 1 > EV s, x s,thatis,fort t 0 τ, t 1, 3.5 holds. Now, we assume that, for t t m1,t m, m 1,,...,, 3.5 holds. For m 1, we will show that 3.5 holds. To this end, we first prove that EV t,x t λ 1 ε. 3.7 In fact, by.1,., C1,and C 3 EV t,x t EV ( t,h ( x ( t ))) λ E λ γ E ( ) } x t τ H ( x ( t λ γ EV ( t,x ( t )) λ 1 )) } τ 3.8 λ 1 ε. Secondly, we assume that there exists s t,t 1, such that 3.7 holds. By 3.7 and the continuity of EV t, x t on t,t 1, we now that there exist t t,t 1, ρ > 0 such that for every t 1, t t, t ρ, t<t, 3.3 and 3.4 hold. Let s inft t,t 1 : EV t, x t λ 1 ε, EV u, x u > λ 1 ε,u t, t ρ t,t 1, EV t 1,x t 1 < EV t,x t, for every t 1, t t, t ρ, t 1 < t }, where ρ > 0is some scalar. Then for t s, s ρ, 3.5 and 3.6 hold. This is a contraction, that is, 3.5 holds for t t,t 1. By mathematical induction, 3.5 holds for any m 1,,..., which implies that system.1 is mean squarely stable.

9 International Differential Equations 9 Remar 3.6. When μ 1, both the continuous dynamics and discrete dynamics are stable under the conditions in Theorem 3.5, so the impulse system can be mean squarely stable regardless of how often or how seldom impulses occur. When μ<1, we have the following results. Theorem 3.7. Assume that there exist scalars λ >λ 1 > 0, λ τ > 0, λ 0, matrix P > 0, and Lyapunov-Krasovsii functional V t, x t V 1,, such that (C1), (C), and (C 3) μ sup N λ λ /λ 1 γ } < 1 hold, then i if 0 < μλ λ τ λ τ ln μ, system.1 is mean squarely stable over impulsive time sequences N sup μ ln μ/ μλ λ τ ; ii if μλ λ τ 0, system.1 is mean squarely stable over any impulsive time sequences. Proof. We prove i and omit the proof of ii. Because μ<1and0<μλ λ τ λ τ ln μ, then there exist a sufficiently small ρ 0 > 0, such that μ ρ 0 < 1, λ ( ) μ ρ 0 λτ > 0, ln μ ln ( ) μ ρ 0 λ μ 1 λ τ λ ( ) 1λτ. μ ρ For any given ε > 0, choose 0 < δ μ ρ 0 λ 1 /λ ε. We assume the initial function ϕ PC b F 0 δ. For t t 0 τ, t 0,by C1, 3.9,andx t0 PC b F 0 δ, weobtain } EV t, x t λ E x t0 τ λ δ ( ) λ 1 μ ρ0 ε <λ 1 ε Now we will prove that 3.5 holds. Otherwise, there exists s t 0,t 1, such that 3.7 holds. Set t inf t t 0,t 1 : EV t, x t λ 1 ε }, 3.31 then by 3.7, 3.30, and the continuity of EV t, x t on t 0,t 1, we now that t t 0,t 1, EV t,x t λ 1 ε.set t sup t t 0,t ( ) } : EV t, x t λ 1 μ ρ0 ε, 3.3 then by 3.30 and the continuity of EV t, x t, we have t t 0,t, EV t, x t λ 1 μ ρ 0 ε and for t t, t, EV t, x t λ 1 ε 1 EV t, x t. ρ 0 μ 3.33

10 10 International Differential Equations Conditions C and C 3 imply that for t t, t, D EV t, x t λev t, x t λ τ EV t, x t ( λ λ τ μ ρ 0 ) EV t, x t By Lemma.1, 3.9, 3.6,andt 1 t 0 ln μ/ λ λ τ /μ, we have ( EV t,x t exp λ ( <exp λ λ τ μ ρ 0 λ τ μ ρ 0 ) ( ) } ( ( )) t t EV t, x t ) } (μ ) t 1 t 0 ρ0 λ1 ε 3.35 λ 1 ε, this is a contradiction with the fact EV t,x t λ 1 ε. Now, we assume that, for t t m1,t m, m 1,,...,, 3.5 holds. For m 1, we will show that 3.5 holds. To this end, we first prove that EV t,x t ( μ ρ 0 ) λ1 ε In fact, by.1,., C1,and C 3 EV t,x t EV ( t,h ( x ( t ))) λ E λ γ E ( ) } x t τ H ( x ( t λ γ EV ( t,x ( t )) λ 1 )) } τ 3.37 ( μ ρ 0 ) λ1 ε. Secondly, we assume that there exists s t,t 1, such that 3.7 holds. Set t inf t t,t 1 : EV t, x t λ 1 ε }, t sup t t,t ( ) } : EV t, x t λ 1 μ ρ0 ε, 3.38 then by 3.37 and the continuity of EV t, x t on t,t 1, we have t t,t 1, t t,t and EV t,x t λ 1 ε, EV t, x t μ ρ 0 λ 1 ε. On the other hand, for t t, t, 3.33 and 3.34 hold, which lead to a contradiction, that is, 3.5 holds for t t,t 1. By mathematical induction, 3.5 holds for any m 1,,..., which implies that system.1 is mean squarely stable.

11 International Differential Equations Application and Numerical Example As an application, we consider the stochastic impulsive Hopfield neural networ with delays in Yang et al. 9 as follows: dx t [ Cx t Af x t Bg x t ] dt σ t, x t,x t dω t, t t 0, t/ t, ( ( x t H x t )), 1,,..., x t 0 θ ϕ θ, s τ, 0, 4.1 where the initial value ϕ s PC b F 0 δ, x t x 1 t,x t,...,x n t T R n is the state vector, C diag c 1,c,...,c n, c i > 0 is the neuron-charging time constant, A a ij n n are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. f x t f 1 x 1 t,f x t,...,f n x n t T R n and g x t g 1 x 1t,g x t,...,g n x nt R n, where f i x i t and g i x it denote, respectively, the measures of response or activation to its incoming potentials of the unit i at time t and time t τ i.we also assume that H 0 0, 1,,..., f 0 0, g 0 0, and σ t, 0, 0 0, then system 4.1 admits an equilibrium solution x t 0. Moreover, we assume that H satisfies., and f, g, σ satisfy f x t Fx t, g xt Gxt, 4. [ ] tr σ T t, x t,x t σ t, x t,x t Kx t K τ x t, 4.3 where F, G, K, andk τ are nown constant matrices with appropriate dimensions. Corollary 4.1. Assume that there exist positive scalars ε 1, ε, β, symmetric matrix P>0and μ sup N λ λ max P /λ min P γ }. Then the following results hold: i if μ>1, λ μλ τ < ln μ/β, then system 4.1 is mean squarely stable over impulsive time sequence N inf τ β ; ii if μ 1, λ λ τ sequence; 0, then system 4.1 is mean squarely stable over any impulsive time iii if μ < 1 and 0 < μλ λ τ λ τ ln μ, then system 4.1 is mean squarely stable over impulsive time sequence N sup μ ln μ/ μλ λ τ ; iv if μ<1, μλ λ τ 0, then system 4.1 is mean squarely stable over any impulsive time sequence, where ( λ λ max C ε 1 AA T P ε BB T P P 1( )) ε 1 1 FT F λ max P K T K, ( λ τ λ max P 1( ε 1 GT G λ max P Kτ T K τ )). 4.4

12 1 International Differential Equations Remar 4.. Obviously, for this application, we extended and improved the according results in Yang et al. 9. By Corollary 4.1, we consider the numerical example in Yang et al. 9. [ ] [ ][ ] [ ][ ] [ ] dx1 t x1 t sin x1 t dx t 0 1. x t arctan x t ( sin x 1 t 1 ) ( ( x 1 t x t 1 ) [ ] arctan x t 1 ) dt 3 dω1 t ( x 1 t 1 ), t t 0, t/ t, x t dω t [ ] x1 t x t [ 0.5 e ][ ( )] x1 t ( ), 1,,..., x t 4.5 where t 0 0. Similar to the result, we can verify that the point 0, 0 T is an equilibrium point and can obtain by calculation that ( ) ( ) P, K T K, and ε i 1 i 1,,λ max P 0.68, λ min P 0.56,γ 0.60 exp 0.1, K T τ K τ I,F G I,μ < 1,λ ,λ τ 3, and, hence, we have μλ λ τ.63, which implies by iv in Corollary 4.1 that the above system is mean squarely stable over any impulsive time sequence. 5. Conclusion In this paper, mean square stability of a class of impulsive stochastic differential equations with time delay has been considered. By Lyapunov-Krasovaii function and stochastic analysis, we obtain some new criteria ensuring mean square stability of the system.1. Some related results in Chen and Zheng 3 and Yang et al. 9 have been improved. Acnowledgment This wor is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.

13 International Differential Equations 13 References 1 V. Lashmiantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood, Chichester, UK, W. H. Chen and W. X. Zheng, Robust stability and H -control of uncertain impulsive systems with time-delay, Automatica, vol. 45, no. 1, pp , Z. G. Li, C. B. Soh, and X. H. Xu, Stability of impulsive differential systems, Mathematical Analysis and Applications, vol. 16, no., pp , Z. G. Li, C. B. Soh, and X. H. Xu, Robust stability of a class of hybrid dynamic uncertain systems, International Robust and Nonlinear Control, vol. 8, no. 1, pp , Z. G. Li, C. Y. Wen, and Y. C. Soh, Analysis and design of impulsive control systems, IEEE Transactions on Automatic Control, vol. 46, no. 6, pp , T. Yang, Impulsive Systems and Control: Theory and Applications, Nova Science, New Yor, NY, USA, B. Autonio and B. Alfonso, Delay-dependent stability of reset systems, Automatica, vol. 46, no. 1, pp. 16 1, J. Yang, S. M. Zhong, and W. P. Luo, Mean square stability analysis of impulsive stochastic differential equations with delays, Computational and Applied Mathematics, vol. 16, no., pp , W. H. Chen, F. Chen, and X. M. Lu, Exponential stability of a class of singularly perturbed stochastic time-delay systems with impulse effect, Nonlinear Analysiis: Real World Applications, vol. 11, no. 5, pp , X. R. Mao, N. Koroleva, and A. Rodina, Robust stability of uncertain stochastic differential delay equations, Systems & Control Letters, vol. 35, no. 5, pp , B. Shi, D. C. Zhang, and M. J. Gai, Theory and Applications of Differential Equations, National Defence Industy Press, Beijing, China, 005.

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