# Mean Square Stability of Impulsive Stochastic Differential Systems

Authors: Shujie Yang; Bao Shi; Mo Li
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

Based on Lyapunov-Krasovskii 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.


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## Body

## 1. Introduction

It is recognized that the theory of impulsive systems provides a natural framework 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 Lakshmikantham et al. [1] and Bainov and Simeonov [2]. 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], Li et al. [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 makes 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 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 knowledge, 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.


## 2. Preliminaries

Throughout this paper, unless explicitly given, for symmetric matricesA 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. λmax(·) (λmin(·)) 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,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC([-τ,0],ℝn) denote the set of piecewise right continuous function ϕ:[-τ,0]→ℝn with the norm defined by ∥ϕ∥τ=sup-τ≤θ≤0∥ϕ(θ)∥, where τ is a known positive constant, PC(δ)={φ∣φ∈PC([-τ,0],ℝn),∥φ∥τ≤δ}, PCℱ0b([-τ,0],ℝn) denote the family of all ℱ0-measurable PC([-τ,0],ℝn)-valued stochastic process φ={φ(s):-τ≤s≤0} with sup-τ≤s≤0𝔼{∥φ(s)∥2}<∞, where 𝔼{·} represents the mathematical expectation operator with respect to the probability measure P, PCℱ0b(δ)={φ∣φ∈PCℱ0b([-τ,0],ℝn),sup-τ≤s≤0𝔼{∥φ(s)∥2}≤δ, ℒ denote the well-known ℒ-operator given by the Itô's formula.In this paper, we consider a class of Itô impulsive stochastic differential systems with time delay(2.1)dx(t)=f(t,x(t),xt)dt+g(t,x(t),xt)dω(t),t≥t0,t≠tk,x(tk)=Hk(x(tk-)),k=1,2,…,x(t0+θ)=φ(θ),θ∈[-τ,0],
where the initial value φ∈PCℱ0b(δ), the fixed impulsive time moments tk satisfy 0≤t0<t1<t2<⋯<tk<⋯(tk→∞ as k→∞). x(t)∈ℝn is the system state, f∈C(ℝ×ℝn×ℝn,ℝn), g∈C(ℝ×ℝn×ℝn,ℝn×m). ω(t)∈ℝm is an standard Brownian motion defined on the complete probability space (Ω,ℱ,{ℱt}t≥0,P). Besides, we assume that Hk(0)=0,(k=1,2,…), f(t,0,0)=0, g(t,0,0)=0 and
(2.2)‖Hk(x(tk-))‖≤γk‖x(tk-)‖,γk≥0,k=1,2,….In the following, we will divide three cases to consider the mean square stability of system (2.1). We denote by 𝒩inf(β) and 𝒩sup(β) the class of impulsive time sequences that satisfy infk{tk-tk-1}≥β and supk{tk-tk-1}≤β, respectively.We need the following lemma and definitions.Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]).
Assume thatf,F∈C(G), g⊂ℝ2 and
(2.3)f(t,x)<F(t,x),(t,x)∈G.
If ϕ(t) (t∈U1) and Φ(t) (t∈U2) are the solutions of Cauchy problems
(2.4)ẋ=f(t,x),(t,x)∈G,x(τ)=ξ,ẋ=F(t,x),(t,x)∈G,x(τ)=ξ,
respectively, then for t∈(τ,∞)∩U1∩U2,
(2.5)ϕ(t)<Φ(t)
and for t∈(-∞,τ)∩U1∩U2,
(2.6)ϕ(t)>Φ(t).Definition 2.2.
For a given class𝒩 of admissible impulsive time sequence, the solution of (2.1) is called mean squarely stable if for any ɛ>0, there exists a scalar δ>0, such that the initial function φ∈PCℱ0b(δ) implies 𝔼{∥x(t)∥2}<ɛ, t≥t0 for all admissible time sequence in 𝒩.Definition 2.3 (see Yang et al. [9]).
The functionV:[t0-τ,∞)×ℝn→ℝ+ belongs to class 𝒱(1,2) if
(1)
the functionV(t,x) is continuously differentiable in t and twice continuously differentiable in x on each of the sets [tk-1,tk)×ℝn,(k=1,2,…) and for all t≥t0, V(t,0)≡0,(2)
V(t,x) is locally Lipschitaian in x,(3)
for eachk=1,2,…, there exist finite limits
(2.7)lim⁡(t,y)→(tk-,x)V(t,y)=V(tk-,x),lim⁡(t,y)→(tk+,x)V(t,y)=V(tk+,x),V(tk+,x)=V(tk,x).


## 3. Main Results

Theorem 3.1.
Assume that there exist scalarsλ2>λ1>0, λτ>0, β>0, λ≤0, ρ>0 matrix P>0 and Lyapunov-Krasovskii functional V(t,x(t))∈𝒱(1,2), such that
(C1)
λ1∥x(t)∥2≤V(t,x(t))≤λ2∥xt∥τ2,(C2)
𝔼ℒV(t,x(t))≤λ𝔼V(t,x(t))+λτ𝔼V(t,xt),t∈[tk-1,tk),k=1,2,…, whenever 𝔼V(t,xt)≤(μ+ρ)𝔼V(t,x(t)),(C3)
μ=supk∈ℕ{λk′=(λ2/λ1)γk2}>1 and λ+(μ+ρ)λτ≤-((ln(μ+ρ))/β),
then the trivial solution of system (2.1) is mean squarely stable over 𝒩inf(τ+β).Proof.
For any givenɛ>0, choose 0<δ≤λ1/(μ+ρ)λ2ɛ. We assume that the initial function φ∈PCℱ0b(δ) and denote the solution x(t,t0,φ) of system (2.1) through (t0,φ) by x(t). In the following, we will prove that x(t) is mean square stable over 𝒩inf(τ+β). For V(t,x(t))∈𝒱(1,2), by Itô formula, for t≠tk,k=1,2,…, we have
(3.1)dV(t,x(t))=LV(t,x(t))dt+Vx(t,x(t))g(t,x(t))dω(t),
where ℒV(t,x(t))=Vt(t,x(t))+Vx(t,x(t))f+(1/2)tr(gTVxxg).
Fort∈[tk-1,tk), k=1,2,…, integrate (3.1) from tk-1 to t, we have
(3.2)V(t,x(t))=V(tk-1,x(tk-1))+∫tk-1tLV(s,x(s))ds+∫tk-1tVx(s,x(s))g(s,x(s))dω(s).
Taking the mathematical expectation of both sides of the above equation, we obtain
(3.3)EV(t,x(t))=EV(tk-1,x(tk-1))+∫tk-1tELV(s,x(s))ds.
So for s∈[t,t+Δt] with t+Δt∈[tk-1,tk) and Δt>0, if 𝔼V(s,xs)≤(μ+ρ)𝔼V(s,x(s)), then we have by (C2)
(3.4)EV(t+Δt,x(t+Δt))-EV(t,x(t))=∫tt+ΔtELV(s,x(s))ds≤∫tt+Δt(λEV(s,x(s))+λτEV(s,xs))ds.
In what follows, we first prove that fort∈[t0-τ,t1),
(3.5)EV(t,x(t))≤λ1ɛ2.
Obviously, for t∈[t0-τ,t0], by (C1) and xt0∈PCℱ0b(δ), we obtain
(3.6)EV(t,x(t))≤λ2E{‖xt0‖τ2}≤λ2δ2≤λ1μ+ρɛ2<λ1ɛ2.
Now it needs only to prove that for t∈(t0,t1), (3.5) holds. Otherwise, there exists s∈(t0,t1), such that
(3.7)EV(s,x(s))>λ1ɛ2.
Set
(3.8)s1=inf⁡{t∈(t0,t1):EV(t,x(t))>λ1ɛ2},
then by (3.6), (3.7), and the continuity of 𝔼V(t,x(t)) on [t0,t1), we know that s1∈(t0,t1),
(3.9)EV(s1,x(s1))=λ1ɛ2,
and for t∈[t0-τ,s1], (3.5) holds. Set
(3.10)s2=sup⁡{t∈[t0,s1):EV(t,x(t))≤λ1μ+ρɛ2},
then by (3.6) and the continuity of 𝔼V(t,x(t)), we have s2∈[t0,s1),
(3.11)EV(s2,x(s2))=λ1μ+ρɛ2,
and for t∈[s2,s1],
(3.12)EV(t,xt)≤λ1ɛ2≤(μ+ρ)EV(t,x(t)),
which implies with (3.4) and (C3) that for t∈[s2,s1],
(3.13)D+EV(t,x(t))≤λEV(t,x(t))+λτEV(t,xt)≤(λ+λτ(μ+ρ))EV(t,x(t))≤0.
This is a contradiction with (3.9) and (3.11).
Now, we assume that, fort∈[tm-1,tm), m=1,2,…,k, (3.5) holds. For m=k+1, we will show that (3.5) holds. To this end, we first prove that for θ∈[-τ,0],
(3.14)EV(tk-+θ,x(tk-+θ))≤λ1μ+ρɛ2.
Noticing [tk-τ,tk)⊂[tk-1,tk), we assume that there exists some s∈[tk-τ,tk], such that
(3.15)EV(s-,x(s-))>λ1μ+ρɛ2,
then there are two cases to be considered.
(i)
For allt∈[tk-1,s], 𝔼V(t-,x(t-))>(λ1/(μ+ρ))ɛ2. Hence, for t∈[tk-1,s], (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1),
(3.16)EV(s-,x(s-))≤exp⁡{(λ+(μ+ρ)λτ)(s-tk-1)}EV(tk-1,x(tk-1))≤λ1ɛ2exp⁡{(λ+(μ+ρ)λτ)(tk-tk-1-τ)}≤λ1μ+ρɛ2,
this is a contradiction with the assumption.(ii)
There exists somet∈[tk-1,s), such that 𝔼V(t,x(t))≤(λ1/(μ+ρ))ɛ2. Set
(3.17)s1=sup⁡{t∈[tk-1,s):EV(t,x(t))≤λ1μ+ρɛ2},
then s1∈[tk-1,s),
(3.18)EV(s1,x(s1))=λ1μ+ρɛ2
and for t∈[s1,s], (3.12) and (3.13) hold, which is a contradiction with (3.15) and (3.18), that is,  (3.14) holds.
By (2.1), (2.2), and (3.14), we have
(3.19)EV(tk,x(tk))=EV(tk,Hk(x(tk-)))≤λ2E{‖Hk(x(tk-))‖τ2}≤λ2γk2E{‖x(tk-)‖τ2}≤λ2γk2λ1sup⁡-τ≤θ≤0EV(tk+θ,x(tk-+θ))≤λk′λ1μ+ρɛ2<λ1ɛ2.
Now we will prove that (3.5) holds for t∈[tk,tk+1). Otherwise, there exists some t∈(tk,tk+1), such that (3.7) holds. Let
(3.20)s1=inf⁡{t∈(tk,tk+1):EV(t,x(t))>λ1ɛ2}.
Then by (3.14), (3.19), and the continuity of 𝔼V(t,x(t)) on [tk,tk+1), we know that s1∈(tk,tk+1) and 𝔼V(s1,x(s1))=λ1ɛ2. If there exists t∈[tk,s1], such that 𝔼V(t,x(t))≤(λ1/μ)ɛ2, then let
(3.21)s2=sup⁡{t∈[tk,s1):EV(t,x(t))≤λ1μ+ρɛ2}.
Otherwise, let s2=tk. Then for t∈[s2,s1], we obtain (3.12) and (3.13), which follows a contradiction.
By mathematical induction, (3.5) holds for any m=1,2,…, which implies that system (2.1) is mean squarely stable.
If substituting condition
(C'1)λ1∥x(t)∥2≤V(t,x(t))≤λ2∥x(t)∥2
for (C1) in Theorem (3.1), then we have the following result.Theorem 3.2.
Assume that there exist scalarsλ2>λ1>0, λτ>0, λ≤0, ρ>0, matrix P>0 and Lyapunov-Krasovskii functional V(t,x(t))∈𝒱(1,2), such that conditions (C'1), (C2), and (C3) hold, then the trivial solution of system (2.1) is mean square stable over 𝒩inf(β).Proof.
The proof is similar to Theorem (3.1), so we omit it. The proof is complete.Remark 3.3.
Comparing the results in Theorems (3.1) and (3.2), we find the influence of the time delay on the mean square stability of system (2.1).Remark 3.4.
Whenμ>1, the impulses which may be destabilizing, so we require the impulses should not happen so frequently.
Whenμ=1, we have the following results.Theorem 3.5.
Assume that there exist scalarsλ2>λ1>0, λτ>0, λ≤0, matrix P>0 and Lyapunov-Krasovskii functional V(t,x(t))∈𝒱(1,2), such that condition (C1) and(C'2)
𝔼ℒV(t,x(t))≤λ𝔼V(t,x(t))+λτ𝔼V(t,xt), t∈[tk-1,tk),k=1,2,… whenever 𝔼V(t,xt)≤𝔼V(t,xt),(C'3)
μ=supk∈ℕ{λk′=(λ2/λ1)γk2}=1, λ+λτ≤0hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.Proof.
For any givenɛ>0, choose 0<δ≤λ1/λ2ɛ. We assume that the initial function φ∈PCℱ0b(δ). In what follows, we first prove that for t≥t0, (3.5) holds.
Obviously, fort∈[t0-τ,t0], by (C1) and xt0∈PCℱ0b(δ), we obtain
(3.22)EV(t,x(t))≤λ2E{‖xt0‖τ2}≤λ2δ2≤λ1ɛ2.
Now we should prove that (3.5) holds. Otherwise, there exists s∈(t0,t1), such that (3.7) holds. By (3.22) and the continuity of 𝔼V(t,x(t)) on [t0,t1), we know there exist t¯∈[t0,t1) and small scalar ρ>0, such that
(3.23)EV(t¯,x(t¯))=λ1ɛ2
and for every t¯1,t¯2∈[t¯,t¯+ρ], t¯1<t¯2,
(3.24)EV(t¯1,x(t¯1))<EV(t¯2,x(t¯2)).
Let s=inf{t∈[t0,t1):𝔼V(t,x(t))=λ1ɛ2,𝔼V(u,x(u))>λ1ɛ2,u∈(t,t+ρ1]⊂[t0,t1),𝔼V(t¯1,x(t¯1))<𝔼V(t¯2,x(t¯2)),foreveryt¯1,t¯2∈[t,t+ρ1],t¯1<t¯2}, where ρ1>0 is some scalar. Then [s,s+ρ1]⊂[t0,t1) and for t∈[s,s+ρ1],
(3.25)EV(t,xt)≤EV(t,x(t)),
which implies with (C'2) and (C'3) that for t∈[s,s+ρ1],
(3.26)D+EV(t,x(t))≤λEV(t,x(t))+λτEV(t,xt)≤(λ+λτ)EV(t,x(t))≤0.
This is a contradiction with the fact 𝔼V(s+ρ1,x(s+ρ1))>𝔼V(s,x(s)), that is, for t∈[t0-τ,t1), (3.5) holds.
Now, we assume that, fort∈[tm-1,tm), m=1,2,…,k, (3.5) holds. For m=k+1, we will show that (3.5) holds. To this end, we first prove that
(3.27)EV(tk,x(tk))≤λ1ɛ2.
In fact, by (2.1), (2.2), (C1), and (C'3)
(3.28)EV(tk,x(tk))=EV(tk,Hk(x(tk-)))≤λ2E{‖Hk(x(tk-))‖τ2}≤λ2γk2E{‖x(tk-)‖τ2}≤λ2γk2λ1EV(tk,x(tk-))≤λ1ɛ2.
Secondly, we assume that there existss∈(tk,tk+1), such that (3.7) holds. By (3.27) and the continuity of 𝔼V(t,x(t)) on [tk,tk+1), we know that there exist t¯∈[tk,tk+1), ρ2>0 such that for every t¯1,t¯2∈[t¯,t¯+ρ2], t¯<t¯2, (3.23) and (3.24) hold.
Lets=inf{t∈[tk,tk+1):𝔼V(t,x(t))=λ1ɛ2,𝔼V(u,x(u))>λ1ɛ2,u∈(t,t+ρ2]⊂[tk,tk+1),𝔼V(t¯1,x(t¯1))<𝔼V(t¯2,x(t¯2)),foreveryt¯1,t¯2∈[t,t+ρ2],t¯1<t¯2}, where ρ2>0 is some scalar.
Then fort∈[s,s+ρ2], (3.25) and (3.26) hold. This is a contraction, that is, (3.5) holds for t∈[tk,tk+1). By mathematical induction, (3.5) holds for any m=1,2,…, which implies that system (2.1) is mean squarely stable.Remark 3.6.
Whenμ=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.
Whenμ<1, we have the following results.Theorem 3.7.
Assume that there exist scalarsλ2>λ1>0, λτ>0, λ≤0, matrix P>0, and Lyapunov-Krasovskii functional V(t,x(t))∈𝒱(1,2), such that (C1), (C2), and
(C”3)μ=supk∈ℕ{λk′=(λ2/λ1)γk2}<1
hold, then
(i)
if0<μλ+λτ≤-λτlnμ, system (2.1) is mean squarely stable over impulsive time sequences 𝒩sup(-μlnμ/(μλ+λτ));(ii)
ifμλ+λτ≤0, system (2.1) is mean squarely stable over any impulsive time sequences.Proof.
We prove (i) and omit the proof of (ii).
Becauseμ<1 and 0<μλ+λτ≤-λτlnμ, then there exist a sufficiently small ρ0>0, such that
(3.29)μ+ρ0<1,λ(μ+ρ0)+λτ>0,-ln⁡μλ+μ-1λτ≤-ln⁡(μ+ρ0)λ+(μ+ρ0)-1λτ.
For any given ɛ>0, choose 0<δ≤((μ+ρ0)λ1)/λ2ɛ. We assume the initial function φ∈PCℱ0b(δ). For t∈[t0-τ,t0], by (C1), (3.29), and xt0∈PCℱ0b(δ), we obtain
(3.30)EV(t,x(t))≤λ2E{‖xt0‖τ2}≤λ2δ2≤λ1(μ+ρ0)ɛ2<λ1ɛ2.
Now we will prove that (3.5) holds. Otherwise, there exists s∈(t0,t1), such that (3.7) holds. Set
(3.31)t*=inf⁡{t∈(t0,t1):EV(t,x(t))≥λ1ɛ2},
then by (3.7), (3.30), and the continuity of 𝔼V(t,x(t)) on [t0,t1), we know that t*∈(t0,t1), 𝔼V(t*,x(t*))=λ1ɛ2. Set
(3.32)t¯=sup⁡{t∈[t0,t*):EV(t,x(t))≤λ1(μ+ρ0)ɛ2},
then by (3.30) and the continuity of 𝔼V(t,x(t)), we have t¯∈[t0,t*), 𝔼V(t¯,x(t¯))=λ1(μ+ρ0)ɛ2 and for t∈[t¯,t*],
(3.33)EV(t,xt)≤λ1ɛ2≤1ρ0+μEV(t,x(t)).
Conditions (C2) and (C”3) imply that for t∈[t¯,t*],
(3.34)D+EV(t,x(t))≤λEV(t,x(t))+λτEV(t,xt)≤(λ+λτμ+ρ0)EV(t,x(t)).
By Lemma (2.1), (3.29), (3.26), and t1-t0≤(-lnμ/(λ+(λτ/μ))), we have
(3.35)EV(t*,x(t*))≤exp⁡{(λ+λτμ+ρ0)(t*-t¯)}EV(t¯,x(t¯))<exp{(λ+λτμ+ρ0)(t1-t0)}(μ+ρ0)λ1ɛ2≤λ1ɛ2,
this is a contradiction with the fact 𝔼V(t*,x(t*))=λ1ɛ2.
Now, we assume that, fort∈[tm-1,tm), m=1,2,…,k, (3.5) holds. For m=k+1, we will show that (3.5) holds. To this end, we first prove that
(3.36)EV(tk,x(tk))≤(μ+ρ0)λ1ɛ2.
In fact, by (2.1), (2.2), (C1), and (C”3)
(3.37)EV(tk,x(tk))=EV(tk,Hk(x(tk-)))≤λ2E{‖Hk(x(tk-))‖τ2}≤λ2γk2E{‖x(tk-)‖τ2}≤λ2γk2λ1EV(tk,x(tk-))≤(μ+ρ0)λ1ɛ2.
Secondly, we assume that there existss∈(tk,tk+1), such that (3.7) holds. Set
(3.38)t*=inf⁡{t∈(tk,tk+1):EV(t,x(t))≥λ1ɛ2},t¯=sup⁡{t∈[tk,t*):EV(t,x(t))≤λ1(μ+ρ0)ɛ2},
then by (3.37) and the continuity of 𝔼V(t,x(t)) on [tk,tk+1), we have t*∈(tk,tk+1), t¯∈[tk,t*) and 𝔼V(t*,x(t*))=λ1ɛ2, 𝔼V(t¯,x(t¯))=(μ+ρ0)λ1ɛ2.
On the other hand, fort∈[t¯,t*], (3.33) and (3.34) hold, which lead to a contradiction, that is, (3.5) holds for t∈[tk,tk+1). By mathematical induction, (3.5) holds for any m=1,2,…, which implies that system (2.1) is mean squarely stable.


## 4. Application and Numerical Example

As an application, we consider the stochastic impulsive Hopfield neural network with delays in Yang et al. [9] as follows:
(4.1)dx(t)=[-Cx(t)+Af(x(t))+Bg(xt)]dt+σ(t,x(t),xt)dω(t),t≥t0,t≠tk,x(tk)=Hk(x(tk-)),k=1,2,…,x(t0+θ)=φ(θ),s∈[-τ,0],
where the initial value φ(s)∈PCℱ0b(δ), x(t)=(x1(t),x2(t),…,xn(t))T∈ℝn is the state vector, C=diag(c1,c2,…,cn), ci>0 is the neuron-charging time constant, A=(aij)n×n are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. f(x(t))=(f1(x1(t)),f2(x2(t)),…,fn(xn(t)))T∈ℝn and g(xt)=(g1(x1t),g2(x2t),…,gn(xnt))ℝn, where fi(xi(t)) and gi(xit) denote, respectively, the measures of response or activation to its incoming potentials of the unit i at time t and time t-τi. We also assume that Hk(0)=0, (k=1,2,…), 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 (2.2), and f(·), g(·), σ(·) satisfy
(4.2)‖f(x(t))‖≤‖Fx(t)‖,‖g(xt)‖≤‖Gxt‖,(4.3)tr⁡[σT(t,x(t),xt)σ(t,x(t),xt)]≤‖Kx(t)‖2+‖Kτxt‖2,
where F, G, K, and Kτ are known constant matrices with appropriate dimensions.Corollary 4.1.
Assume that there exist positive scalarsɛ1, ɛ2, β, symmetric matrix P>0 and μ=supk∈ℕ{λ′k=(λmax(P)/λmin(P))γk2}. Then the following results hold:
(i)
ifμ>1, λ+μλτ<-lnμ/β, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩inf(τ+β);(ii)
ifμ=1, λ+λτ≤0, then system (4.1) is mean squarely stable over any impulsive time sequence;(iii)
ifμ<1 and 0<μλ+λτ≤-λτlnμ, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩sup(-μlnμ/(μλ+λτ));(iv)
ifμ<1, μλ+λτ≤0, then system (4.1) is mean squarely stable over any impulsive time sequence, where
(4.4)λ=λmax⁡(-2C+ɛ1AATP+ɛ2BBTP+P-1(ɛ1-1FTF+λmax⁡(P)KTK)),λτ=λmax⁡(P-1(ɛ2-1GTG+λmax⁡(P)KτTKτ)).Remark 4.2.
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]. 							(4.5)[dx1(t)dx2(t)]={[sinx1(t-12)arctanx2(t-12)][-10.500-12.2][x1(t)x2(t)]+[1.2-0.20.62.4][sinx1(t)arctanx2(t)]+[1.60.3-0.51.8]×[sinx1(t-12)arctanx2(t-12)]}dt+[2x1(t)x2(t-13)x1(t-12)-x2(t)][dω1(t)dω2(t)],t≥t0,t≠tk,[x1(tk)x2(tk)]=e-0.1k[0.5-0.150.120.6][x1(tk-)x2(tk-)],k=1,2,…,
where  t0=0.
Similar to the result, we can verify that the point(0,0)T is an equilibrium point and can obtain by calculation that
(4.6)P=(0.56000.68),KTK=(4001),
and ɛi=1(i=1,2),λmax(P)=0.68,λmin(P)=0.56,γk=0.620exp(-0.1k),KτTKτ=I,F=G=I,μ=0.4668<1,λ=-12.0443,λτ=3, and, hence, we have μλ+λτ=-2.6223, 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-Krasovakii function and stochastic analysis, we obtain some new criteria ensuring mean square stability of the system (2.1). Some related results in Chen and Zheng [3] and Yang et al. [9] have been improved.



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*Source: 613695-2011-07-26.xml*