^{1,2}

^{1}

^{3}

^{3}

^{1}

^{2}

^{3}

This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both the

Impulsive dynamical systems have attracted considerable interest in science and engineering in recent years because they provide a natural framework for mathematical modeling of many real-world problems where the reactions undergo abrupt changes [

Stability is one of the most important issues in the study of impulsive stochastic differential systems (see, e.g., [

The average impulsive interval was proposed in [

In this paper, by using the average impulsive interval, we investigate the

The rest of this paper is organized as follows. In Section

Throughout this paper, let

For

In this paper, we consider the following impulsive stochastic delay differential systems:

As a standing hypothesis,

Let

The purpose of this paper is to discuss the stability of system (

The trivial solution of system (

almost exponentially stable if the solution

The average impulsive interval of the impulsive sequence

In this section, we will establish some stability criteria of stochastic delay differential system with destabilizing impulses or stabilizing impulses. The first theorem addresses the case where the continuous dynamics in the system (

Assume that there exist positive constants

for

there exists a positive constant

According to (H_{1}), we see that
_{3}), we get (_{2}), we have

Theorem

In Theorem

In the following theorem, when the continuous dynamics in the system (

Let

for

there exists a positive constant

In view of (H_{1}), we obtain
_{2}) and (

According to (

Theorem

In Theorem

Let

for

there exists a positive constant

The proof is similar to the proof given in Theorem

The following theorem shows that the trivial solution of system (

Assume that

By Theorem

In this section, two numerical examples are given to show the effectiveness of the main results derived in the preceding section.

Consider an impulsive stochastic delay differential system as follows:

Figure

Impulses of Example

Impulsive disturbance of Example

Consider an impulsive stochastic delay differential system as follows:

The stabilizing impulsive sequence in the system (

Impulses of Example

Impulsive control of Example

The

The authors would like to thank the Associate Editor and anonymous referee for their helpful comments and suggestions which greatly improved this paper. This work was supported by the National Natural Science Foundation of China (no. 10871041 and no. 71171003). Also, this work was partially supported by the Natural Science Foundation of Anhui Province (no. 10040606Q03).