#### 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 and , the notion (, , ) means is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. () represents the maximum (minimum) eigenvalue of the corresponding matrix, respectively. denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let () be a complete probability space with a filtration satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let denote the set of piecewise right continuous function with the norm defined by , where is a known positive constant, , denote the family of all -measurable -valued stochastic process with , where represents the mathematical expectation operator with respect to the probability measure , , denote the well-known -operator given by the ItΓ΄'s formula.

In this paper, we consider a class of ItΓ΄ impulsive stochastic differential systems with time delay where the initial value , the fixed impulsive time moments satisfy as . is the system state, , . is an standard Brownian motion defined on the complete probability space . Besides, we assume that , , and

In the following, we will divide three cases to consider the mean square stability of system (2.1). We denote by and the class of impulsive time sequences that satisfy and , respectively.

We need the following lemma and definitions.

Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]). *Assume that , and
**
If () and () are the solutions of Cauchy problems
**
respectively, then for ,
**
and for , β
*

*Definition 2.2. *For a given class of admissible impulsive time sequence, the solution of (2.1) is called mean squarely stable if for any , there exists a scalar , such that the initial function implies , for all admissible time sequence in .

*Definition 2.3 (see Yang et al. [9]). *The function belongs to class if

(1)the function is continuously differentiable in and twice continuously differentiable in on each of the sets and for all , , (2) is locally Lipschitaian in , (3)for each , there exist finite limits

#### 3. Main Results

Theorem 3.1. *Assume that there exist scalars , , , , matrix and Lyapunov-Krasovskii functional , such that *

(C1)*, *(C2)*, whenever ,*(C3)* and ,**then the trivial solution of system (2.1) is mean squarely stable over .*

*Proof. *For any given , choose . We assume that the initial function and denote the solution of system (2.1) through by . In the following, we will prove that is mean square stable over . For , by ItΓ΄ formula, for , we have
where .

For , , integrate (3.1) from to , we have
Taking the mathematical expectation of both sides of the above equation, we obtain
So for with and , if , then we have by (C2)

In what follows, we first prove that for ,
Obviously, for , by (C1) and , we obtain
Now it needs only to prove that for , (3.5) holds. Otherwise, there exists , such that
Set
then by (3.6), (3.7), and the continuity of on , we know that ,β
and for , (3.5) holds. Set
then by (3.6) and the continuity of , we have ,
and for ,
which implies with (3.4) and (C3) that for ,
This is a contradiction with (3.9) and (3.11).

Now, we assume that, for , , (3.5) holds. For , we will show that (3.5) holds. To this end, we first prove that for ,
Noticing , we assume that there exists some , such that
then there are two cases to be considered.

(i)For all , . Hence, for , (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1),
this is a contradiction with the assumption.(ii)There exists some , such that . Set
then ,
and for , (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

Now we will prove that (3.5) holds for . Otherwise, there exists some , such that (3.7) holds. Let
Then by (3.14), (3.19), and the continuity of on , we know that and . If there exists , such that , then let
Otherwise, let . Then for , we obtain (3.12) and (3.13), which follows a contradiction.

By mathematical induction, (3.5) holds for any , which implies that system (2.1) is mean squarely stable.

If substituting condition

(C'1)

for (C1) in Theorem (3.1), then we have the following result.

Theorem 3.2. *Assume that there exist scalars , , , , matrix and Lyapunov-Krasovskii functional , such that conditions (C'1), (C2), and (C3) hold, then the trivial solution of system (2.1) is mean square stable over .*

* 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 , the impulses which may be destabilizing, so we require the impulses should not happen so frequently.

When , we have the following results.

Theorem 3.5. *Assume that there exist scalars , , , matrix and Lyapunov-Krasovskii functional , such that condition (C1) and *(C'2)*, whenever ,*(C'3)*, **hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.*

* Proof. *For any given , choose . We assume that the initial function . In what follows, we first prove that for , (3.5) holds.

Obviously, for , by (C1) and , we obtain
Now we should prove that (3.5) holds. Otherwise, there exists , such that (3.7) holds. By (3.22) and the continuity of on , we know there exist and small scalar , such that
and for every , ,
Let , where is some scalar. Then and for ,
which implies with (C'2) and (C'3) that for ,
This is a contradiction with the fact , that is, for , (3.5) holds.

Now, we assume that, for , , (3.5) holds. For , we will show that (3.5) holds. To this end, we first prove that
In fact, by (2.1), (2.2), (C1), and (C'3)

Secondly, we assume that there exists , such that (3.7) holds. By (3.27) and the continuity of on , we know that there exist , such that for every , , (3.23) and (3.24) hold.

Let , where is some scalar.

Then for , (3.25) and (3.26) hold. This is a contraction, that is, (3.5) holds for . By mathematical induction, (3.5) holds for any , which implies that system (2.1) is mean squarely stable.

*Remark 3.6. *When , 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 , we have the following results.

Theorem 3.7. *Assume that there exist scalars , , , matrix , and Lyapunov-Krasovskii functional , such that (C1), (C2), and **(Cβ3) **
hold, then *

(i)*if , system (2.1) is mean squarely stable over impulsive time sequences ; *(ii)*if , system (2.1) is mean squarely stable over any impulsive time sequences.*

*Proof. *We prove (i) and omit the proof of (ii).

Because and , then there exist a sufficiently small , such that
For any given , choose . We assume the initial function . For , by (C1), (3.29), and , we obtain

Now we will prove that (3.5) holds. Otherwise, there exists , such that (3.7) holds. Set
then by (3.7), (3.30), and the continuity of on , we know that , . Set
then by (3.30) and the continuity of , we have , and for ,
Conditions (C2) and (Cβ3) imply that for ,
By Lemma (2.1), (3.29), (3.26), and , we have
this is a contradiction with the fact .

Now, we assume that, for , , (3.5) holds. For , we will show that (3.5) holds. To this end, we first prove that
In fact, by (2.1), (2.2), (C1), and (Cβ3)

Secondly, we assume that there exists , such that (3.7) holds. Set
then by (3.37) and the continuity of on , we have , and , .

On the other hand, for , (3.33) and (3.34) hold, which lead to a contradiction, that is, (3.5) holds for . By mathematical induction, (3.5) holds for any , 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: where the initial value , is the state vector, , is the neuron-charging time constant, are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. and , where and denote, respectively, the measures of response or activation to its incoming potentials of the unit at time and time . We also assume that , , , , and , then system (4.1) admits an equilibrium solution . Moreover, we assume that satisfies (2.2), and , , satisfy where , , , and are known constant matrices with appropriate dimensions.

Corollary 4.1. *Assume that there exist positive scalars , , , symmetric matrix and . Then the following results hold: *

(i)*if , , then system (4.1) is mean squarely stable over impulsive time sequence ;*(ii)*if , , then system (4.1) is mean squarely stable over any impulsive time sequence;*(iii)*if and , then system (4.1) is mean squarely stable over impulsive time sequence ;*(iv)*if , , then system (4.1) is mean squarely stable over any impulsive time sequence, where
*

*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].
where .

Similar to the result, we can verify that the point is an equilibrium point and can obtain by calculation that
and , and, hence, we have , 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.