International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 613695 | 13 pages | https://doi.org/10.1155/2011/613695

Mean Square Stability of Impulsive Stochastic Differential Systems

Academic Editor: Xingfu Zou
Received04 Dec 2010
Revised17 May 2011
Accepted26 May 2011
Published26 Jul 2011

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.

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 matrices 𝐴 and 𝐡, the notion 𝐴β‰₯𝐡 (𝐴>𝐡, 𝐴≀𝐡, 𝐴<𝐡) means π΄βˆ’π΅ is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. πœ†max(β‹…) (πœ†min(β‹…)) represents the maximum (minimum) eigenvalue of the corresponding matrix, respectively. β€–β‹…β€– denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let (Ξ©,β„±,{ℱ𝑑}𝑑β‰₯0,𝑃) be a complete probability space with a filtration {ℱ𝑑}𝑑β‰₯0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC([βˆ’πœ,0],ℝ𝑛) denote the set of piecewise right continuous function πœ™βˆΆ[βˆ’πœ,0]→ℝ𝑛 with the norm defined by β€–πœ™β€–πœ=supβˆ’πœβ‰€πœƒβ‰€0β€–πœ™(πœƒ)β€–, where 𝜏 is a known positive constant, PC(𝛿)={πœ‘βˆ£πœ‘βˆˆPC([βˆ’πœ,0],ℝ𝑛),β€–πœ‘β€–πœβ‰€π›Ώ}, PC𝑏ℱ0([βˆ’πœ,0],ℝ𝑛) denote the family of all β„±0-measurable PC([βˆ’πœ,0],ℝ𝑛)-valued stochastic process πœ‘={πœ‘(𝑠)βˆΆβˆ’πœβ‰€π‘ β‰€0} with supβˆ’πœβ‰€π‘ β‰€0𝔼{β€–πœ‘(𝑠)β€–2}<∞, where 𝔼{β‹…} represents the mathematical expectation operator with respect to the probability measure 𝑃, PC𝑏ℱ0(𝛿)={πœ‘βˆ£πœ‘βˆˆPC𝑏ℱ0([βˆ’πœ,0],ℝ𝑛),supβˆ’πœβ‰€π‘ β‰€0𝔼{β€–πœ‘(𝑠)β€–2}≀𝛿, β„’ denote the well-known β„’-operator given by the ItΓ΄'s formula.

In this paper, we consider a class of ItΓ΄ impulsive stochastic differential systems with time delay ξ€·dπ‘₯(𝑑)=𝑓𝑑,π‘₯(𝑑),π‘₯𝑑d𝑑+𝑔𝑑,π‘₯(𝑑),π‘₯𝑑dπœ”(𝑑),𝑑β‰₯𝑑0,π‘‘β‰ π‘‘π‘˜,π‘₯ξ€·π‘‘π‘˜ξ€Έ=π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜π‘₯𝑑,π‘˜=1,2,…,0ξ€Έ[],+πœƒ=πœ‘(πœƒ),πœƒβˆˆβˆ’πœ,0(2.1) where the initial value πœ‘βˆˆPC𝑏ℱ0(𝛿), the fixed impulsive time moments π‘‘π‘˜ satisfy 0≀𝑑0<𝑑1<𝑑2<β‹―<π‘‘π‘˜<β‹―(π‘‘π‘˜β†’βˆž as π‘˜β†’βˆž). π‘₯(𝑑)βˆˆβ„π‘› is the system state, π‘“βˆˆπΆ(ℝ×ℝ𝑛×ℝ𝑛,ℝ𝑛), π‘”βˆˆπΆ(ℝ×ℝ𝑛×ℝ𝑛,β„π‘›Γ—π‘š). πœ”(𝑑)βˆˆβ„π‘š is an standard Brownian motion defined on the complete probability space (Ξ©,β„±,{ℱ𝑑}𝑑β‰₯0,𝑃). Besides, we assume that π»π‘˜(0)=0,(π‘˜=1,2,…), 𝑓(𝑑,0,0)=0, 𝑔(𝑑,0,0)=0 and β€–β€–π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜β€–β€–ξ€Έξ€Έβ‰€π›Ύπ‘˜β€–β€–π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έβ€–β€–,π›Ύπ‘˜β‰₯0,π‘˜=1,2,….(2.2)

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 infπ‘˜{π‘‘π‘˜βˆ’π‘‘π‘˜βˆ’1}β‰₯𝛽 and supπ‘˜{π‘‘π‘˜βˆ’π‘‘π‘˜βˆ’1}≀𝛽, respectively.

We need the following lemma and definitions.

Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]). Assume that 𝑓,𝐹∈𝐢(𝐺), π‘”βŠ‚β„2 and 𝑓(𝑑,π‘₯)<𝐹(𝑑,π‘₯),(𝑑,π‘₯)∈𝐺.(2.3) If πœ™(𝑑) (π‘‘βˆˆπ‘ˆ1) and Ξ¦(𝑑) (π‘‘βˆˆπ‘ˆ2) are the solutions of Cauchy problems π‘₯Μ‡π‘₯=𝑓(𝑑,π‘₯),(𝑑,π‘₯)∈𝐺,(𝜏)=πœ‰,Μ‡π‘₯=𝐹(𝑑,π‘₯),(𝑑,π‘₯)∈𝐺,π‘₯(𝜏)=πœ‰,(2.4) respectively, then for π‘‘βˆˆ(𝜏,∞)βˆ©π‘ˆ1βˆ©π‘ˆ2, πœ™(𝑑)<Ξ¦(𝑑)(2.5) and for π‘‘βˆˆ(βˆ’βˆž,𝜏)βˆ©π‘ˆ1βˆ©π‘ˆ2,   πœ™(𝑑)>Ξ¦(𝑑).(2.6)

Definition 2.2. For a given class 𝒩 of admissible impulsive time sequence, the solution of (2.1) is called mean squarely stable if for any πœ€>0, there exists a scalar 𝛿>0, such that the initial function πœ‘βˆˆPC𝑏ℱ0(𝛿) implies 𝔼{β€–π‘₯(𝑑)β€–2}<πœ€, 𝑑β‰₯𝑑0 for all admissible time sequence in 𝒩.

Definition 2.3 (see Yang et al. [9]). The function π‘‰βˆΆ[𝑑0βˆ’πœ,∞)×ℝ𝑛→ℝ+ belongs to class 𝒱(1,2) if
(1)the function 𝑉(𝑑,π‘₯) is continuously differentiable in 𝑑 and twice continuously differentiable in π‘₯ on each of the sets [π‘‘π‘˜βˆ’1,π‘‘π‘˜)×ℝ𝑛,(π‘˜=1,2,…) and for all 𝑑β‰₯𝑑0, 𝑉(𝑑,0)≑0, (2)𝑉(𝑑,π‘₯) is locally Lipschitaian in π‘₯, (3)for each π‘˜=1,2,…, there exist finite limits lim𝑑(𝑑,𝑦)β†’βˆ’π‘˜ξ€Έ,π‘₯𝑉𝑑(𝑑,𝑦)=π‘‰βˆ’π‘˜ξ€Έ,π‘₯,lim𝑑(𝑑,𝑦)β†’+π‘˜ξ€Έ,π‘₯𝑉𝑑(𝑑,𝑦)=𝑉+π‘˜ξ€Έ,𝑉𝑑,π‘₯+π‘˜ξ€Έξ€·π‘‘,π‘₯=π‘‰π‘˜ξ€Έ.,π‘₯(2.7)

3. Main Results

Theorem 3.1. Assume that there exist scalars πœ†2>πœ†1>0, πœ†πœ>0, 𝛽>0, πœ†β‰€0, 𝜌>0 matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’±(1,2), such that
(C1)πœ†1β€–π‘₯(𝑑)β€–2≀𝑉(𝑑,π‘₯(𝑑))β‰€πœ†2β€–π‘₯𝑑‖2𝜏, (C2)𝔼ℒ𝑉(𝑑,π‘₯(𝑑))β‰€πœ†π”Όπ‘‰(𝑑,π‘₯(𝑑))+πœ†πœπ”Όπ‘‰(𝑑,π‘₯𝑑),π‘‘βˆˆ[π‘‘π‘˜βˆ’1,π‘‘π‘˜),π‘˜=1,2,…, whenever 𝔼𝑉(𝑑,π‘₯𝑑)≀(πœ‡+𝜌)𝔼𝑉(𝑑,π‘₯(𝑑)),(C3)πœ‡=supπ‘˜βˆˆβ„•{πœ†ξ…žπ‘˜=(πœ†2/πœ†1)𝛾2π‘˜}>1 and πœ†+(πœ‡+𝜌)πœ†πœβ‰€βˆ’((ln(πœ‡+𝜌))/𝛽),
then the trivial solution of system (2.1) is mean squarely stable over 𝒩inf(𝜏+𝛽).

Proof. For any given πœ€>0, choose √0<π›Ώβ‰€πœ†1/(πœ‡+𝜌)πœ†2πœ€. We assume that the initial function πœ‘βˆˆPC𝑏ℱ0(𝛿) and denote the solution π‘₯(𝑑,𝑑0,πœ‘) of system (2.1) through (𝑑0,πœ‘) by π‘₯(𝑑). In the following, we will prove that π‘₯(𝑑) is mean square stable over 𝒩inf(𝜏+𝛽). For 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’±(1,2), by ItΓ΄ formula, for π‘‘β‰ π‘‘π‘˜,π‘˜=1,2,…, we have d𝑉(𝑑,π‘₯(𝑑))=ℒ𝑉(𝑑,π‘₯(𝑑))d𝑑+𝑉π‘₯(𝑑,π‘₯(𝑑))𝑔(𝑑,π‘₯(𝑑))dπœ”(𝑑),(3.1) where ℒ𝑉(𝑑,π‘₯(𝑑))=𝑉𝑑(𝑑,π‘₯(𝑑))+𝑉π‘₯(𝑑,π‘₯(𝑑))𝑓+(1/2)tr(𝑔𝑇𝑉π‘₯π‘₯𝑔).
For π‘‘βˆˆ[π‘‘π‘˜βˆ’1,π‘‘π‘˜), π‘˜=1,2,…, integrate (3.1) from π‘‘π‘˜βˆ’1 to 𝑑, we have 𝑑𝑉(𝑑,π‘₯(𝑑))=π‘‰π‘˜βˆ’1𝑑,π‘₯π‘˜βˆ’1+ξ€œξ€Έξ€Έπ‘‘π‘‘π‘˜βˆ’1ξ€œβ„’π‘‰(𝑠,π‘₯(𝑠))d𝑠+π‘‘π‘‘π‘˜βˆ’1𝑉π‘₯(𝑠,π‘₯(𝑠))𝑔(𝑠,π‘₯(𝑠))dπœ”(𝑠).(3.2) Taking the mathematical expectation of both sides of the above equation, we obtain 𝑑𝔼𝑉(𝑑,π‘₯(𝑑))=π”Όπ‘‰π‘˜βˆ’1𝑑,π‘₯π‘˜βˆ’1+ξ€œξ€Έξ€Έπ‘‘π‘‘π‘˜βˆ’1𝔼ℒ𝑉(𝑠,π‘₯(𝑠))d𝑠.(3.3) So for π‘ βˆˆ[𝑑,𝑑+Δ𝑑] with 𝑑+Ξ”π‘‘βˆˆ[π‘‘π‘˜βˆ’1,π‘‘π‘˜) and Δ𝑑>0, if 𝔼𝑉(𝑠,π‘₯𝑠)≀(πœ‡+𝜌)𝔼𝑉(𝑠,π‘₯(𝑠)), then we have by (C2) ξ€œπ”Όπ‘‰(𝑑+Δ𝑑,π‘₯(𝑑+Δ𝑑))βˆ’π”Όπ‘‰(𝑑,π‘₯(𝑑))=𝑑𝑑+Ξ”π‘‘β‰€ξ€œπ”Όβ„’π‘‰(𝑠,π‘₯(𝑠))d𝑠𝑑𝑑+Ξ”π‘‘ξ€·πœ†π”Όπ‘‰(𝑠,π‘₯(𝑠))+πœ†πœξ€·π”Όπ‘‰π‘ ,π‘₯𝑠d𝑠.(3.4)
In what follows, we first prove that for π‘‘βˆˆ[𝑑0βˆ’πœ,𝑑1), 𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†1πœ€2.(3.5) Obviously, for π‘‘βˆˆ[𝑑0βˆ’πœ,𝑑0], by (C1) and π‘₯𝑑0∈PC𝑏ℱ0(𝛿), we obtain 𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†2𝔼‖‖π‘₯𝑑0β€–β€–2πœξ‚‡β‰€πœ†2𝛿2β‰€πœ†1πœ€πœ‡+𝜌2<πœ†1πœ€2.(3.6) Now it needs only to prove that for π‘‘βˆˆ(𝑑0,𝑑1), (3.5) holds. Otherwise, there exists π‘ βˆˆ(𝑑0,𝑑1), such that 𝔼𝑉(𝑠,π‘₯(𝑠))>πœ†1πœ€2.(3.7) Set 𝑠1𝑑=infπ‘‘βˆˆ0,𝑑1ξ€ΈβˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))>πœ†1πœ€2ξ€Ύ,(3.8) then by (3.6), (3.7), and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)) on [𝑑0,𝑑1), we know that 𝑠1∈(𝑑0,𝑑1),  𝑠𝔼𝑉1𝑠,π‘₯1ξ€Έξ€Έ=πœ†1πœ€2,(3.9) and for π‘‘βˆˆ[𝑑0βˆ’πœ,𝑠1], (3.5) holds. Set 𝑠2𝑑=supπ‘‘βˆˆ0,𝑠1ξ€Έπœ†βˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))≀1πœ€πœ‡+𝜌2ξ‚Ό,(3.10) then by (3.6) and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)), we have 𝑠2∈[𝑑0,𝑠1), 𝑠𝔼𝑉2𝑠,π‘₯2=πœ†ξ€Έξ€Έ1πœ€πœ‡+𝜌2,(3.11) and for π‘‘βˆˆ[𝑠2,𝑠1], 𝔼𝑉𝑑,π‘₯π‘‘ξ€Έβ‰€πœ†1πœ€2≀(πœ‡+𝜌)𝔼𝑉(𝑑,π‘₯(𝑑)),(3.12) which implies with (3.4) and (C3) that for π‘‘βˆˆ[𝑠2,𝑠1], 𝐷+𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†π”Όπ‘‰(𝑑,π‘₯(𝑑))+πœ†πœξ€·π”Όπ‘‰π‘‘,π‘₯π‘‘ξ€Έβ‰€ξ€·πœ†+πœ†πœξ€Έ(πœ‡+𝜌)𝔼𝑉(𝑑,π‘₯(𝑑))≀0.(3.13) This is a contradiction with (3.9) and (3.11).
Now, we assume that, for π‘‘βˆˆ[π‘‘π‘šβˆ’1,π‘‘π‘š), π‘š=1,2,…,π‘˜, (3.5) holds. For π‘š=π‘˜+1, we will show that (3.5) holds. To this end, we first prove that for πœƒβˆˆ[βˆ’πœ,0], ξ€·π‘‘π”Όπ‘‰βˆ’π‘˜ξ€·π‘‘+πœƒ,π‘₯βˆ’π‘˜β‰€πœ†+πœƒξ€Έξ€Έ1πœ€πœ‡+𝜌2.(3.14) Noticing [π‘‘π‘˜βˆ’πœ,π‘‘π‘˜)βŠ‚[π‘‘π‘˜βˆ’1,π‘‘π‘˜), we assume that there exists some π‘ βˆˆ[π‘‘π‘˜βˆ’πœ,π‘‘π‘˜], such that 𝔼𝑉(π‘ βˆ’,π‘₯(π‘ βˆ’πœ†))>1πœ€πœ‡+𝜌2,(3.15) then there are two cases to be considered.
(i)For all π‘‘βˆˆ[π‘‘π‘˜βˆ’1,𝑠], 𝔼𝑉(π‘‘βˆ’,π‘₯(π‘‘βˆ’))>(πœ†1/(πœ‡+𝜌))πœ€2. Hence, for π‘‘βˆˆ[π‘‘π‘˜βˆ’1,𝑠], (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1), 𝔼𝑉(π‘ βˆ’,π‘₯(π‘ βˆ’))≀expξ€½ξ€·πœ†+(πœ‡+𝜌)πœ†πœξ€Έξ€·π‘ βˆ’π‘‘π‘˜βˆ’1ξ€·π‘‘ξ€Έξ€Ύπ”Όπ‘‰π‘˜βˆ’1𝑑,π‘₯π‘˜βˆ’1ξ€Έξ€Έβ‰€πœ†1πœ€2expξ€½ξ€·πœ†+(πœ‡+𝜌)πœ†πœπ‘‘ξ€Έξ€·π‘˜βˆ’π‘‘π‘˜βˆ’1β‰€πœ†βˆ’πœξ€Έξ€Ύ1πœ€πœ‡+𝜌2,(3.16) this is a contradiction with the assumption.(ii)There exists some π‘‘βˆˆ[π‘‘π‘˜βˆ’1,𝑠), such that 𝔼𝑉(𝑑,π‘₯(𝑑))≀(πœ†1/(πœ‡+𝜌))πœ€2. Set 𝑠1𝑑=supπ‘‘βˆˆπ‘˜βˆ’1ξ€Έπœ†,π‘ βˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))≀1πœ€πœ‡+𝜌2ξ‚Ό,(3.17) then 𝑠1∈[π‘‘π‘˜βˆ’1,𝑠), 𝑠𝔼𝑉1𝑠,π‘₯1=πœ†ξ€Έξ€Έ1πœ€πœ‡+𝜌2(3.18) and for π‘‘βˆˆ[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 ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯π‘˜ξ€·π‘‘ξ€Έξ€Έ=π”Όπ‘‰π‘˜,π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έξ€Έξ€Έβ‰€πœ†2π”Όξ‚†β€–β€–π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜β€–β€–ξ€Έξ€Έ2πœξ‚‡β‰€πœ†2𝛾2π‘˜π”Όξ‚†β€–β€–π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έβ€–β€–2πœξ‚‡β‰€πœ†2𝛾2π‘˜πœ†1supβˆ’πœβ‰€πœƒβ‰€0ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘+πœƒ,π‘₯βˆ’π‘˜+πœƒξ€Έξ€Έβ‰€πœ†ξ…žπ‘˜πœ†1πœ€πœ‡+𝜌2<πœ†1πœ€2.(3.19)
Now we will prove that (3.5) holds for π‘‘βˆˆ[π‘‘π‘˜,π‘‘π‘˜+1). Otherwise, there exists some π‘‘βˆˆ(π‘‘π‘˜,π‘‘π‘˜+1), such that (3.7) holds. Let 𝑠1𝑑=infπ‘‘βˆˆπ‘˜,π‘‘π‘˜+1ξ€ΈβˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))>πœ†1πœ€2ξ€Ύ.(3.20) Then by (3.14), (3.19), and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)) on [π‘‘π‘˜,π‘‘π‘˜+1), we know that 𝑠1∈(π‘‘π‘˜,π‘‘π‘˜+1) and 𝔼𝑉(𝑠1,π‘₯(𝑠1))=πœ†1πœ€2. If there exists π‘‘βˆˆ[π‘‘π‘˜,𝑠1], such that 𝔼𝑉(𝑑,π‘₯(𝑑))≀(πœ†1/πœ‡)πœ€2, then let 𝑠2𝑑=supπ‘‘βˆˆπ‘˜,𝑠1ξ€Έπœ†βˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))≀1πœ€πœ‡+𝜌2ξ‚Ό.(3.21) Otherwise, let 𝑠2=π‘‘π‘˜. Then for π‘‘βˆˆ[𝑠2,𝑠1], we obtain (3.12) and (3.13), which follows a contradiction.
By mathematical induction, (3.5) holds for any π‘š=1,2,…, which implies that system (2.1) is mean squarely stable.
If substituting condition
(C'1) πœ†1β€–π‘₯(𝑑)β€–2≀𝑉(𝑑,π‘₯(𝑑))β‰€πœ†2β€–π‘₯(𝑑)β€–2
for (C1) in Theorem (3.1), then we have the following result.

Theorem 3.2. Assume that there exist scalars πœ†2>πœ†1>0, πœ†πœ>0, πœ†β‰€0, 𝜌>0, matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’±(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. When πœ‡>1, the impulses which may be destabilizing, so we require the impulses should not happen so frequently.
When πœ‡=1, we have the following results.

Theorem 3.5. Assume that there exist scalars πœ†2>πœ†1>0, πœ†πœ>0, πœ†β‰€0, matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’±(1,2), such that condition (C1) and (C'2)𝔼ℒ𝑉(𝑑,π‘₯(𝑑))β‰€πœ†π”Όπ‘‰(𝑑,π‘₯(𝑑))+πœ†πœπ”Όπ‘‰(𝑑,π‘₯𝑑), π‘‘βˆˆ[π‘‘π‘˜βˆ’1,π‘‘π‘˜),π‘˜=1,2,… whenever 𝔼𝑉(𝑑,π‘₯𝑑)≀𝔼𝑉(𝑑,π‘₯𝑑),(C'3)πœ‡=supπ‘˜βˆˆβ„•{πœ†ξ…žπ‘˜=(πœ†2/πœ†1)𝛾2π‘˜}=1, πœ†+πœ†πœβ‰€0hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.

Proof. For any given πœ€>0, choose √0<π›Ώβ‰€πœ†1/πœ†2πœ€. We assume that the initial function πœ‘βˆˆPC𝑏ℱ0(𝛿). In what follows, we first prove that for 𝑑β‰₯𝑑0, (3.5) holds.
Obviously, for π‘‘βˆˆ[𝑑0βˆ’πœ,𝑑0], by (C1) and π‘₯𝑑0∈PC𝑏ℱ0(𝛿), we obtain 𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†2𝔼‖‖π‘₯𝑑0β€–β€–2πœξ‚‡β‰€πœ†2𝛿2β‰€πœ†1πœ€2.(3.22) Now we should prove that (3.5) holds. Otherwise, there exists π‘ βˆˆ(𝑑0,𝑑1), such that (3.7) holds. By (3.22) and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)) on [𝑑0,𝑑1), we know there exist π‘‘βˆˆ[𝑑0,𝑑1) and small scalar 𝜌>0, such that 𝔼𝑉𝑑,π‘₯𝑑=πœ†1πœ€2(3.23) and for every 𝑑1,𝑑2∈[𝑑,𝑑+𝜌], 𝑑1<𝑑2, 𝔼𝑉𝑑1ξ€·,π‘₯𝑑1ξ€·ξ€Έξ€Έ<𝔼𝑉𝑑2ξ€·,π‘₯𝑑2.ξ€Έξ€Έ(3.24) Let 𝑠=inf{π‘‘βˆˆ[𝑑0,𝑑1)βˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))=πœ†1πœ€2,𝔼𝑉(𝑒,π‘₯(𝑒))>πœ†1πœ€2,π‘’βˆˆ(𝑑,𝑑+𝜌1]βŠ‚[𝑑0,𝑑1),𝔼𝑉(𝑑1,π‘₯(𝑑1))<𝔼𝑉(𝑑2,π‘₯(𝑑2)),forevery𝑑1,𝑑2∈[𝑑,𝑑+𝜌1],𝑑1<𝑑2}, where 𝜌1>0 is some scalar. Then [𝑠,𝑠+𝜌1]βŠ‚[𝑑0,t1) and for π‘‘βˆˆ[𝑠,𝑠+𝜌1], 𝔼𝑉𝑑,π‘₯𝑑≀𝔼𝑉(𝑑,π‘₯(𝑑)),(3.25) which implies with (C'2) and (C'3) that for π‘‘βˆˆ[𝑠,𝑠+𝜌1], 𝐷+𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†π”Όπ‘‰(𝑑,π‘₯(𝑑))+πœ†πœξ€·π”Όπ‘‰π‘‘,π‘₯π‘‘ξ€Έβ‰€ξ€·πœ†+πœ†πœξ€Έπ”Όπ‘‰(𝑑,π‘₯(𝑑))≀0.(3.26) This is a contradiction with the fact 𝔼𝑉(𝑠+𝜌1,π‘₯(𝑠+𝜌1))>𝔼𝑉(𝑠,π‘₯(𝑠)), that is, for π‘‘βˆˆ[𝑑0βˆ’πœ,𝑑1), (3.5) holds.
Now, we assume that, for π‘‘βˆˆ[π‘‘π‘šβˆ’1,π‘‘π‘š), π‘š=1,2,…,π‘˜, (3.5) holds. For π‘š=π‘˜+1, we will show that (3.5) holds. To this end, we first prove that ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯π‘˜ξ€Έξ€Έβ‰€πœ†1πœ€2.(3.27) In fact, by (2.1), (2.2), (C1), and (C'3) ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯π‘˜ξ€·π‘‘ξ€Έξ€Έ=π”Όπ‘‰π‘˜,π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έξ€Έξ€Έβ‰€πœ†2π”Όξ‚†β€–β€–π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜β€–β€–ξ€Έξ€Έ2πœξ‚‡β‰€πœ†2𝛾2π‘˜π”Όξ‚†β€–β€–π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έβ€–β€–2πœξ‚‡β‰€πœ†2𝛾2π‘˜πœ†1ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯βˆ’π‘˜ξ€Έξ€Έβ‰€πœ†1πœ€2.(3.28)
Secondly, we assume that there exists π‘ βˆˆ(π‘‘π‘˜,π‘‘π‘˜+1), such that (3.7) holds. By (3.27) and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)) on [π‘‘π‘˜,π‘‘π‘˜+1), we know that there exist π‘‘βˆˆ[π‘‘π‘˜,π‘‘π‘˜+1), 𝜌2>0 such that for every 𝑑1,𝑑2∈[𝑑,𝑑+𝜌2], 𝑑<𝑑2, (3.23) and (3.24) hold.
Let 𝑠=inf{π‘‘βˆˆ[π‘‘π‘˜,π‘‘π‘˜+1)βˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))=πœ†1πœ€2,𝔼𝑉(𝑒,π‘₯(𝑒))>πœ†1πœ€2,π‘’βˆˆ(𝑑,𝑑+𝜌2]βŠ‚[π‘‘π‘˜,π‘‘π‘˜+1),𝔼𝑉(𝑑1,π‘₯(𝑑1))<𝔼𝑉(𝑑2,π‘₯(𝑑2)),forevery𝑑1,𝑑2∈[𝑑,𝑑+𝜌2],𝑑1<𝑑2}, where 𝜌2>0 is some scalar.
Then for π‘‘βˆˆ[𝑠,𝑠+𝜌2], (3.25) and (3.26) hold. This is a contraction, that is, (3.5) holds for π‘‘βˆˆ[π‘‘π‘˜,π‘‘π‘˜+1). By mathematical induction, (3.5) holds for any π‘š=1,2,…, which implies that system (2.1) is mean squarely stable.

Remark 3.6. When πœ‡=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 πœ‡<1, we have the following results.

Theorem 3.7. Assume that there exist scalars πœ†2>πœ†1>0, πœ†πœ>0, πœ†β‰€0, matrix 𝑃>0, and Lyapunov-Krasovskii functional 𝑉(𝑑,π‘₯(𝑑))βˆˆπ’±(1,2), such that (C1), (C2), and
(C”3) πœ‡=supπ‘˜βˆˆβ„•{πœ†ξ…žπ‘˜=(πœ†2/πœ†1)𝛾2π‘˜}<1
hold, then
(i)if 0<πœ‡πœ†+πœ†πœβ‰€βˆ’πœ†πœlnπœ‡, system (2.1) is mean squarely stable over impulsive time sequences 𝒩sup(βˆ’πœ‡lnπœ‡/(πœ‡πœ†+πœ†πœ)); (ii)if πœ‡πœ†+πœ†πœβ‰€0, system (2.1) is mean squarely stable over any impulsive time sequences.

Proof. We prove (i) and omit the proof of (ii).
Because πœ‡<1 and 0<πœ‡πœ†+πœ†πœβ‰€βˆ’πœ†πœlnπœ‡, then there exist a sufficiently small 𝜌0>0, such that πœ‡+𝜌0ξ€·<1,πœ†πœ‡+𝜌0ξ€Έ+πœ†πœ>0,βˆ’lnπœ‡πœ†+πœ‡βˆ’1πœ†πœβ‰€ξ€·βˆ’lnπœ‡+𝜌0ξ€Έξ€·πœ†+πœ‡+𝜌0ξ€Έβˆ’1πœ†πœ.(3.29) For any given πœ€>0, choose √0<𝛿≀((πœ‡+𝜌0)πœ†1)/πœ†2πœ€. We assume the initial function πœ‘βˆˆPC𝑏ℱ0(𝛿). For π‘‘βˆˆ[𝑑0βˆ’πœ,𝑑0], by (C1), (3.29), and π‘₯𝑑0∈PC𝑏ℱ0(𝛿), we obtain 𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†2𝔼‖‖π‘₯𝑑0β€–β€–2πœξ‚‡β‰€πœ†2𝛿2β‰€πœ†1ξ€·πœ‡+𝜌0ξ€Έπœ€2<πœ†1πœ€2.(3.30)
Now we will prove that (3.5) holds. Otherwise, there exists π‘ βˆˆ(𝑑0,𝑑1), such that (3.7) holds. Set π‘‘βˆ—ξ€½ξ€·π‘‘=infπ‘‘βˆˆ0,𝑑1ξ€ΈβˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))β‰₯πœ†1πœ€2ξ€Ύ,(3.31) then by (3.7), (3.30), and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)) on [𝑑0,𝑑1), we know that π‘‘βˆ—βˆˆ(𝑑0,𝑑1), 𝔼𝑉(π‘‘βˆ—,π‘₯(π‘‘βˆ—))=πœ†1πœ€2. Set 𝑑𝑑=supπ‘‘βˆˆ0,π‘‘βˆ—ξ€ΈβˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))β‰€πœ†1ξ€·πœ‡+𝜌0ξ€Έπœ€2ξ€Ύ,(3.32) then by (3.30) and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)), we have π‘‘βˆˆ[𝑑0,π‘‘βˆ—), 𝔼𝑉(𝑑,π‘₯(𝑑))=πœ†1(πœ‡+𝜌0)πœ€2 and for π‘‘βˆˆ[𝑑,π‘‘βˆ—], 𝔼𝑉𝑑,π‘₯π‘‘ξ€Έβ‰€πœ†1πœ€2≀1𝜌0+πœ‡π”Όπ‘‰(𝑑,π‘₯(𝑑)).(3.33) Conditions (C2) and (C”3) imply that for π‘‘βˆˆ[𝑑,π‘‘βˆ—], 𝐷+𝔼𝑉(𝑑,π‘₯(𝑑))β‰€πœ†π”Όπ‘‰(𝑑,π‘₯(𝑑))+πœ†πœξ€·π”Όπ‘‰π‘‘,π‘₯π‘‘ξ€Έβ‰€ξ‚΅πœ†πœ†+πœπœ‡+𝜌0𝔼𝑉(𝑑,π‘₯(𝑑)).(3.34) By Lemma (2.1), (3.29), (3.26), and 𝑑1βˆ’π‘‘0≀(βˆ’lnπœ‡/(πœ†+(πœ†πœ/πœ‡))), we have ξ€·π‘‘π”Όπ‘‰βˆ—ξ€·π‘‘,π‘₯βˆ—πœ†ξ€Έξ€Έβ‰€expξ‚»ξ‚΅πœ†+πœπœ‡+𝜌0ξ‚Άξ€·π‘‘βˆ—βˆ’π‘‘ξ€Έξ‚Όξ€·π”Όπ‘‰ξ€·π‘‘,π‘₯π‘‘πœ†ξ€Έξ€Έ<𝑒π‘₯π‘ξ‚»ξ‚΅πœ†+πœπœ‡+𝜌0𝑑1βˆ’π‘‘0ξ€Έξ‚Όξ€·πœ‡+𝜌0ξ€Έπœ†1πœ€2β‰€πœ†1πœ€2,(3.35) this is a contradiction with the fact 𝔼𝑉(π‘‘βˆ—,π‘₯(π‘‘βˆ—))=πœ†1πœ€2.
Now, we assume that, for π‘‘βˆˆ[π‘‘π‘šβˆ’1,π‘‘π‘š), π‘š=1,2,…,π‘˜, (3.5) holds. For π‘š=π‘˜+1, we will show that (3.5) holds. To this end, we first prove that ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯π‘˜β‰€ξ€·ξ€Έξ€Έπœ‡+𝜌0ξ€Έπœ†1πœ€2.(3.36) In fact, by (2.1), (2.2), (C1), and (C”3) ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯π‘˜ξ€·π‘‘ξ€Έξ€Έ=π”Όπ‘‰π‘˜,π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έξ€Έξ€Έβ‰€πœ†2π”Όξ‚†β€–β€–π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜β€–β€–ξ€Έξ€Έ2πœξ‚‡β‰€πœ†2𝛾2π‘˜π”Όξ‚†β€–β€–π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έβ€–β€–2πœξ‚‡β‰€πœ†2𝛾2π‘˜πœ†1ξ€·π‘‘π”Όπ‘‰π‘˜ξ€·π‘‘,π‘₯βˆ’π‘˜β‰€ξ€·ξ€Έξ€Έπœ‡+𝜌0ξ€Έπœ†1πœ€2.(3.37)
Secondly, we assume that there exists π‘ βˆˆ(π‘‘π‘˜,π‘‘π‘˜+1), such that (3.7) holds. Set π‘‘βˆ—ξ€½ξ€·π‘‘=infπ‘‘βˆˆπ‘˜,π‘‘π‘˜+1ξ€ΈβˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))β‰₯πœ†1πœ€2ξ€Ύ,𝑑𝑑=supπ‘‘βˆˆπ‘˜,π‘‘βˆ—ξ€ΈβˆΆπ”Όπ‘‰(𝑑,π‘₯(𝑑))β‰€πœ†1ξ€·πœ‡+𝜌0ξ€Έπœ€2ξ€Ύ,(3.38) then by (3.37) and the continuity of 𝔼𝑉(𝑑,π‘₯(𝑑)) on [π‘‘π‘˜,π‘‘π‘˜+1), we have π‘‘βˆ—βˆˆ(π‘‘π‘˜,π‘‘π‘˜+1), π‘‘βˆˆ[π‘‘π‘˜,π‘‘βˆ—) and 𝔼𝑉(π‘‘βˆ—,π‘₯(π‘‘βˆ—))=πœ†1πœ€2, 𝔼𝑉(𝑑,π‘₯(𝑑))=(πœ‡+𝜌0)πœ†1πœ€2.
On the other hand, for π‘‘βˆˆ[𝑑,π‘‘βˆ—], (3.33) and (3.34) hold, which lead to a contradiction, that is, (3.5) holds for π‘‘βˆˆ[π‘‘π‘˜,π‘‘π‘˜+1). By mathematical induction, (3.5) holds for any π‘š=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: ξ€Ίξ€·π‘₯dπ‘₯(𝑑)=βˆ’πΆπ‘₯(𝑑)+𝐴𝑓(π‘₯(𝑑))+𝐡𝑔𝑑d𝑑+πœŽπ‘‘,π‘₯(𝑑),π‘₯𝑑dπœ”(𝑑),𝑑β‰₯𝑑0,π‘‘β‰ π‘‘π‘˜,π‘₯ξ€·π‘‘π‘˜ξ€Έ=π»π‘˜ξ€·π‘₯ξ€·π‘‘βˆ’π‘˜π‘₯𝑑,π‘˜=1,2,…,0ξ€Έ[],+πœƒ=πœ‘(πœƒ),π‘ βˆˆβˆ’πœ,0(4.1) where the initial value πœ‘(𝑠)∈PC𝑏ℱ0(𝛿), π‘₯(𝑑)=(π‘₯1(𝑑),π‘₯2(𝑑),…,π‘₯𝑛(𝑑))π‘‡βˆˆβ„π‘› is the state vector, 𝐢=diag(𝑐1,𝑐2,…,𝑐𝑛), 𝑐𝑖>0 is the neuron-charging time constant, 𝐴=(π‘Žπ‘–π‘—)𝑛×𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. 𝑓(π‘₯(𝑑))=(𝑓1(π‘₯1(𝑑)),𝑓2(π‘₯2(𝑑)),…,𝑓𝑛(π‘₯𝑛(𝑑)))π‘‡βˆˆβ„π‘› and 𝑔(π‘₯𝑑)=(𝑔1(π‘₯1𝑑),𝑔2(π‘₯2𝑑),…,𝑔𝑛(π‘₯𝑛𝑑))ℝ𝑛, where 𝑓𝑖(π‘₯𝑖(𝑑)) and 𝑔𝑖(π‘₯𝑖𝑑) denote, respectively, the measures of response or activation to its incoming potentials of the unit 𝑖 at time 𝑑 and time π‘‘βˆ’πœπ‘–. We also assume that π»π‘˜(0)=0, (π‘˜=1,2,…), 𝑓(0)=0, 𝑔(0)=0, and 𝜎(𝑑,0,0)=0, then system (4.1) admits an equilibrium solution π‘₯(𝑑)≑0. Moreover, we assume that 𝐻(β‹…) satisfies (2.2), and 𝑓(β‹…), 𝑔(β‹…), 𝜎(β‹…) satisfy ‖‖𝑔π‘₯‖𝑓(π‘₯(𝑑))‖≀‖𝐹π‘₯(𝑑)β€–,𝑑‖‖≀‖‖𝐺π‘₯π‘‘β€–β€–ξ€ΊπœŽ,(4.2)tr𝑇𝑑,π‘₯(𝑑),π‘₯π‘‘ξ€ΈπœŽξ€·π‘‘,π‘₯(𝑑),π‘₯𝑑≀‖‖𝐾π‘₯(𝑑)2+β€–β€–πΎπœπ‘₯𝑑‖‖2,(4.3) where 𝐹, 𝐺, 𝐾, and 𝐾𝜏 are known constant matrices with appropriate dimensions.

Corollary 4.1. Assume that there exist positive scalars πœ€1, πœ€2, 𝛽, symmetric matrix 𝑃>0 and πœ‡=supπ‘˜βˆˆβ„•{πœ†ξ…žπ‘˜=(πœ†max(𝑃)/πœ†min(𝑃))𝛾2π‘˜}. Then the following results hold:
(i)if πœ‡>1, πœ†+πœ‡πœ†πœ<βˆ’lnπœ‡/𝛽, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩inf(𝜏+𝛽);(ii)if πœ‡=1, πœ†+πœ†πœβ‰€0, then system (4.1) is mean squarely stable over any impulsive time sequence;(iii)if πœ‡<1 and 0<πœ‡πœ†+πœ†πœβ‰€βˆ’πœ†πœlnπœ‡, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩sup(βˆ’πœ‡lnπœ‡/(πœ‡πœ†+πœ†πœ));(iv)if πœ‡<1, πœ‡πœ†+πœ†πœβ‰€0, then system (4.1) is mean squarely stable over any impulsive time sequence, where πœ†=πœ†maxξ€·βˆ’2𝐢+πœ€1𝐴𝐴𝑇𝑃+πœ€2𝐡𝐡𝑇𝑃+π‘ƒβˆ’1ξ€·πœ€1βˆ’1𝐹𝑇𝐹+πœ†max(𝑃)𝐾𝑇𝐾,πœ†ξ€Έξ€Έπœ=πœ†maxξ€·π‘ƒβˆ’1ξ€·πœ€2βˆ’1𝐺𝑇𝐺+πœ†max(𝑃)πΎπ‘‡πœπΎπœ.ξ€Έξ€Έ(4.4)

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]. dπ‘₯1(𝑑)dπ‘₯2(ξƒ­=⎧βŽͺ⎨βŽͺβŽ©ξƒ¬π‘₯𝑑)βˆ’10.500βˆ’12.21π‘₯(𝑑)2(ξƒ­+𝑑)1.2βˆ’0.20.62.4sinπ‘₯1(𝑑)arctanπ‘₯2(ξƒ­+ξƒ¬ξƒ­Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£π‘‘)1.60.3βˆ’0.51.8sinπ‘₯1ξ‚€1π‘‘βˆ’2arctanπ‘₯2ξ‚€1π‘‘βˆ’2ξ‚βŽ€βŽ₯βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺ⎭⎑⎒⎒⎒⎣d𝑑+2π‘₯1(𝑑)π‘₯2ξ‚€1π‘‘βˆ’3π‘₯1ξ‚€1π‘‘βˆ’2ξ‚βˆ’π‘₯2⎀βŽ₯βŽ₯βŽ₯βŽ¦ξƒ¬(𝑑)dπœ”1(𝑑)dπœ”2(𝑑),𝑑β‰₯𝑑0,π‘‘β‰ π‘‘π‘˜,π‘₯1ξ€·π‘‘π‘˜ξ€Έπ‘₯2ξ€·π‘‘π‘˜ξ€Έξƒ­=π‘’βˆ’0.1π‘˜ξƒ¬π‘₯0.5βˆ’0.150.120.61ξ€·π‘‘βˆ’π‘˜ξ€Έπ‘₯2ξ€·π‘‘βˆ’π‘˜ξ€Έξƒ­,π‘˜=1,2,…,(4.5) where 𝑑0=0.
Similar to the result, we can verify that the point (0,0)𝑇 is an equilibrium point and can obtain by calculation that ξƒͺ𝑃=0.56000.68,𝐾𝑇ξƒͺ𝐾=4001,(4.6) and πœ€π‘–=1(𝑖=1,2),πœ†max(𝑃)=0.68,πœ†min(𝑃)=0.56,π›Ύπ‘˜=0.620exp(βˆ’0.1π‘˜),πΎπ‘‡πœπΎπœ=𝐼,𝐹=𝐺=𝐼,πœ‡=0.4668<1,πœ†=βˆ’12.0443,πœ†πœ=3, and, hence, we have πœ‡πœ†+πœ†πœ=βˆ’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.

Acknowledgment

This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.

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