#### Abstract

This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both the th moment and almost sure exponential stability criteria are established by using the average impulsive interval. When the impulses are regarded as disturbances, a lower bound of average impulsive interval is obtained; it means that the impulses should not happen too frequently. On the other hand, when the impulses are used to stabilize the system, an upper bound of average impulsive interval is derived; namely, enough impulses are needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.

#### 1. Introduction

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 [1–3]. These systems have been found to have important applications in various fields, such as control systems with communication constraints [4], network system [5, 6], sampled-data systems [7, 8], and mechanical systems [9]. On the other hand, impulsive control based on impulsive systems can provide an efficient way to deal with plants that cannot endure continuous control inputs [3]. In recent years, the impulsive control theory has been generalized from deterministic systems to stochastic systems and has been shown to have extensive applications [10, 11].

Stability is one of the most important issues in the study of impulsive stochastic differential systems (see, e.g., [12–20]). When the continuous dynamical system is unstable, there is some literature that is concerned with the th moment exponential stability with stabilizing impulses. For example, several criteria on the global exponential stability and instability are obtained in [17]. The th moment exponential stability is discussed in [18] by using the vector Lyapunov functions. The authors in [19] investigated impulsive stabilization of stochastic delay differential systems, and both th moment and almost sure exponential stability criteria are established by using the Lyapunov-Razumikhin method. Recently, both continuous dynamical stable system and continuous dynamical unstable system are studied in [20].

The average impulsive interval was proposed in [21], and it is useful to study the synchronization problem of dynamical networks with destabilizing impulses (see, e.g., [21–23]). The average impulsive interval can be used to control frequency of the impulsive occurrence. When the continuous dynamical system is stable and the impulsive effects are destabilizing, in order to maintain the stability of the system, the impulses should not happen too frequently. Therefore, there should exist a lower bound; if the average impulsive interval is not less than the bound, the stability can be maintained. On the other hand, when the continuous dynamical system is unstable, and the impulses are used to stabilize the unstable system, there should exist enough impulses to stabilize the system, that is, the frequency of impulsive occurrence should exceed a lower bound. Thus there exists an upper bound of the average impulsive interval; if the average interval is less than the upper bound, the system is stabilized by the impulses.

In this paper, by using the average impulsive interval, we investigate the th moment and almost sure exponential stability for stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. When the continuous dynamical system is stable, the lower bound of the average impulsive interval is obtained, by which we can estimate how intensive impulsive disturbance the stable system can endure. On the other hand, when the continuous dynamical system is unstable, the upper bound of the average impulsive interval is derived. From this bound, we can estimate the minimum impulsive frequency needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.

The rest of this paper is organized as follows. In Section 2, we introduce some notations and definitions. We establish several stability criteria for impulsive stochastic delay differential systems in Section 3. In Section 4, two examples are given to illustrate the effectiveness of our results.

#### 2. Preliminaries

Throughout this paper, let be a complete probability space with some filtration satisfying the usual conditions (i.e., the filtration is increasing and right continuous while contains all -null sets). Let be an -dimensional -adapted Brownian motion.

For , denotes the Euclidean norm of . For , we say that a function from to is piecewise continuous if the function has at most a finite number of jumps discontinuous on and continuous from the right for all points in . Given , denotes the family of piecewise continuous functions from to with norm . For and , let be the family of -adapted and -valued random variables such that . Let and .

In this paper, we consider the following impulsive stochastic delay differential systems: where is a strictly increasing sequence such that as and . is defined by , . The mappings , , and are all Borel-measurable functions.

As a standing hypothesis, , , and are assumed to satisfy necessary assumptions so that, for any , system (2.1) has a unique global solution . In addition, we suppose that , , and for all . Then system (2.1) admits a trivial solution .

Let denote the family of all nonnegative functions on that are continuously twice differentiable in and once in . For each , define an operator for system (2.1) by where

The purpose of this paper is to discuss the stability of system (2.1). Let us begin with the following definitions.

*Definition 2.1. *The trivial solution of system (2.1) is said to be(1)th moment exponentially stable if for any initial data , the solution satisfies
where and are positive constants independent of ,(2)almost exponentially stable if the solution satisfies
for any initial data and .

*Definition 2.2. *The average impulsive interval of the impulsive sequence is equal to a positive number if there exists a positive integer such that
where denotes the number of impulsive times of the impulsive sequence on the interval .

#### 3. Main Results

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 (2.1) is stable. It is shown that under some conditions the impulsive disturbance do not destroy the stability of system (2.1).

Theorem 3.1. *Assume that there exist positive constants such that**,** for , ,
provided satisfying that , ,**there exists a positive constant such that
**, .** Then the trivial solution of system (2.1) is th moment exponentially stable.*

*Proof. *According to (H_{1}), we see that
where . We will show that
Suppose (3.4) is not true. Then there exist some such that . Set . It follows that and . Moreover, there is a sequence and such that
Consequently,
which implies that
Noticing that the solution and functionals , are continuous on , thus we obtain
for sufficiently small . Using Itô’s formula, we derive
which yields that
This contradicts (3.5). Therefore (3.4) holds. Now, we assume that
We will prove that
In view of (3.11) and (H_{3}), we get (3.11) holds for . Suppose (3.12) is not true; then, there exist some such that . Seting , we have and . Moreover, there is a sequence and such that
For , there exists an integer such that . Hence
Thus, from (H_{2}), we have
Similarly, this can lead to a contradiction, which implies that (3.12) holds. From Definition 2.2, we see that
Consequently,
where . This completes the proof.

*Remark 3.2. *Theorem 3.1 gives the conditions under which the impulsive disturbances do not destroy the stability of system (2.1). When the impulsive effects are destabilizing, the impulses should not happen too frequently. Therefore, in order to maintain the stability of continuous dynamical system, the average impulsive interval is used to control the impulsive frequency.

*Remark 3.3. *In Theorem 3.1, the impulses are regarded as disturbance; therefore, the condition is reasonable. It is worth pointing out that in Theorem 3.1, for arbitrary small and any , the impulsive interval can be less than and simultaneously the average impulsive intervals are not less than . That is, high-density impulses are allowed to happen in a certain interval, but we need low-density impulses to follow as a compensation.

In the following theorem, when the continuous dynamics in the system (2.1) is unstable, it is shown that the system (2.1) can be stabilized by impulses.

Theorem 3.4. *Let and there is a positive integer such that . Assume that there exist positive constants such that**,** for , ,
provided satisfying that , ,** there exists a positive constant such that
**.** Then the trivial solution of system (2.1) is th moment exponentially stable.*

*Proof. *In view of (H_{1}), we obtain
where . We will show that
Suppose (3.21) is not true. Then there exist some such that . Set , which yields that for and . Moreover, there is a sequence and such that
Noticing that
we derive
Since the solution and functionals , are continuous on , we see that
for sufficiently small . Using Itô’s formula, we obtain
which implies
This contradicts (3.22). Thus (3.21) holds. Now, we assume that
We will prove that
Using (H_{2}) and (3.28) implies that (3.29) holds for . Suppose (3.29) is not true. Then, there exist some such that . Seting , we have and . Moreover, there is a sequence and such that
For , there exists an integer such that , ; then,
It follows that
Similarly, this can lead to a contradiction, which implies that (3.29) holds.

According to (3.16), we obtain
where . This completes the proof.

*Remark 3.5. *Theorem 3.4 shows that an unstable stochastic delay differential system can be successfully stabilized by impulses. The average impulsive interval is used to estimate the impulsive frequency; namely, the impulsive frequency should exceed a lower bound so that there exist enough impulses to stabilize the unstable continuous dynamical system.

In Theorem 3.4, we need to assume that and , which means the impulsive interval cannot be small enough. However, if system (2.1) is an impulsive stochastic differential system without delay, then the system can still be exponential stability when is extremely small.

Corollary 3.6. *Let and in system (2.1). Assume that there exist positive constants such that**,
** for , ,
** there exists a positive constant such that
**.**Then the trivial solution of system (2.1) is th moment exponentially stable.*

*Proof. *The proof is similar to the proof given in Theorem 3.4, so we omit the detailed proof.

The following theorem shows that the trivial solution of system (2.1) is almost surely exponentially stable, under some additional conditions.

Theorem 3.7. *Assume that , and there exists a positive integer such that . Suppose that the conditions in Theorem 3.1 or Theorem 3.4 hold. Moreover, there exists a constant , such that
**
Then the trivial solution of system (2.1) is almost surely exponentially stable.*

*Proof. *By Theorem 3.1 or Theorem 3.4, we derive that the trivial solution of system (2.1) is th moment exponentially stable. Therefore, there exists a positive constant such that
It is obvious that
Combining the Hölder inequality with (3.36) and (3.37) implies that
By virtue of the Burkholder-Davis-Gundy inequality, (3.36), and (3.37), we have
where is a positive constant depending on only. Thanks to (3.36) and (3.37), we see that
Substituting (3.39)–(3.41) into (3.38) gives that
where is a positive constant. Then for an arbitrary and , we derive
Using the Borel-Cantelly lemma, we see that there exists an such that for almost all , ,
where . It follows that
Consequently,
Let . Then the result follows.

#### 4. Numerical Examples

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

*Example 4.1. *Consider an impulsive stochastic delay differential system as follows:
Choosing , , , and in Theorem 3.1, then we have
Seting , then . It is clear that , , . For all , we let , , . Thus, by Theorem 3.1 the trivial solution of system (4.1) is th moment exponential stability. Set , which yields in Theorem 3.7. Obviously, for system (4.1), condition (3.36) holds. Then by Theorem 3.7, the trivial solution of system (4.1) is also almost surely exponential stability.

Figure 1 describes the destabilizing impulsive sequence in the system (4.1) when . It can be seen from Figure 2 that the destabilizing impulses do not destroy the stability of system (4.1).

*Example 4.2. *Consider an impulsive stochastic delay differential system as follows:
Clearly, for system (4.3), condition (3.36) holds. Let , , , and in Theorem 3.4. Then we have
Seting , then . It follows that , . Thus, we can choose , , , which follows , , and . Then by Theorems 3.4 and 3.7, the trivial solution of system (4.3) is th moment and almost sure exponential stability.

The stabilizing impulsive sequence in the system (4.3) is described in Figure 3. It can be seen from Figure 4 that unstable continuous dynamics in the system (4.3) can be successfully stabilized by the impulses.

#### 5. Conclusion

The th moment and almost sure exponential stability are investigated in this paper. By using the average impulsive interval, several sufficient conditions are established for stability of stochastic delay differential systems with destabilizing impulses or stabilizing impulses. Finally, two numerical simulation examples are offered to verify the effectiveness of the main results.

#### Acknowledgments

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).