## Functional Differential and Difference Equations with Applications

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# Exponential Stability of Impulsive Stochastic Functional Differential Systems

**Academic Editor:**Josef Diblík

#### Abstract

This paper is concerned with stabilization of impulsive stochastic delay differential systems. Based on the Razumikhin techniques and Lyapunov functions, several criteria on *p*th moment and almost sure exponential stability are established. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses.

#### 1. Introduction

In the past decades, many authors have obtained various results of deterministic functional differential systems (see [1–6] and the references therein). But it is well known that there are many stochastic factors in the realistic environment, and it is necessary to consider stochastic models. In fact, stochastic functional differential systems (SFDSs) have received more attention in recent years. The properties of SFDSs including stability have been studied in [7–10], which can be widely used in science and engineering (see [11] and the references therein). Furthermore, besides stochastic effects, impulsive effects likewise exist in many evolution processes in which system states change abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics, and telecommunications, and so forth. The impulsive control theory comes to play an important role in science and industry [12]. So the stability investigation of impulsive stochastic differential systems (ISDSs) and impulsive stochastic functional differential systems (ISFDSs) is interesting to many authors [13–20].

Recently, the Razumikhin-type asymptotical stability theorems for ISFESs were established [21, 22]. However, little work has been done on generally exponential stability of ISFESs [23, 24]. In this paper, stability criteria for impulsive stochastic function differential systems are investigated by Razumikhin technique and Lyapunov functions. It is shown that an unstable stochastic delay system can be successfully stabilized by impulses and the results can be easily applied.

#### 2. Preliminaries

Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). means a -dimensional Brownian motion defined on this probability space. denotes the set of real numbers, is the set of nonnegative real numbers, and denotes the -dimensional real space equipped with Euclidean norm . If is a vector or matrix, its transpose is denoted by and its operator norm is denoted by . Moreover, let and denote by the family of continuous functions from to . Let denote the set of positive integers, that is, .

For , a function from to is called piecewise continuous, if the function has at most a finite number of jump discontinuities on , which is continuous from the right for all points in . Given , denotes the family of piecewise continuous functions from to . A norm on is defined as for .

For and , let denote the family of all -measurable -value random variables such that and , denote the family of -value random variables that are bounded and -measurable.

In this paper, we consider the following ISFDS: where the initial value is regarded as a -value process and . Similarly, is defined by and . Both are Borel measurable, and represents the impulsive perturbation of at time . The fixed moments of impulse times satisfy , . Moreover, , , and are assumed to satisfy necessary assumptions so that, for any initial data , system (2.1) has a unique global solution, denoted by (e.g., see [25] for existence and uniqueness results for general impulsive hybrid stochastic delay systems including (2.1)). For the purpose of stability in this note, we also assume the , and for all , , then system (2.1) admits a trivial solution.

*Definition 2.1. *The trivial solution of system (2.1) is said to be th moment exponentially stable if there is a pair of positive constants such that
for all . When , it is usually said to be exponentially stable in mean square. It follows from (2.2) that
The left-hand side of (2.3) is called the th moment Lyapunov exponent of the solution.

*Definition 2.2. *The trivial solution of system (2.1) is said to be almost exponentially stable if there is a pair of positive constants , such that for
for all . It follows from (2.4) that
The left-hand side of (2.5) is called the Lyapunov exponent of the solution.

*Definition 2.3. *Let denote the family of all nonnegative functions on that are continuously twice differential in and once in . If , define the operator for system (2.1) by
where .

#### 3. Main Results

In this section, we will establish some criteria on the th moment exponential stability and almost exponential stability for system (2.1) by using the Razumikhin technique and Lyapunov functions. We begin with the following lemma, which concerns with the continuity of .

Lemma 3.1. *Let , and let be a solution of system (2.1). If there exists such that , then is continuous on .*

*Proof. *By the Itô formula,
for all , where . Since , we can find an integer such that . For any integer , define the stopping time
where as usual. Since is continuous on is also continuous on . Clearly, a.s. as . Moreover, it has , following from . It then follows from the definition of above that
where . So, letting , by the dominated convergence theorem and Fubini’s theorem, we have
for . This implies that is continuous on .

Theorem 3.2. *Let and be a piecewise continuous function. Suppose there exist some positive constants , and such that** for all ,
**for all , and ,
where ,**for all and ,
whenever
where .*

Then the trivial solution of system (2.1) is th moment exponentially stable and its th moment Lyapunov exponent is not greater than .

*Proof. *Given any initial data , the global solution of (2.1) is written as in this proof. Without loss of generality, assume that the initial date is nontrivial so that is not a trivial solution. Choose such that
Then it follows from condition (i) and (3.9) that
In the following, we will show that
In order to do so, we first prove that
If (3.12) is not true, then there exist some such that . Set . Then and also, by the continuity of ) (see Lemma 3.1),
In view of (3.10), define . Then and, by the continuity of ,
Now in view of (3.9), (3.13), and (3.14), one has, for and ,
By the Razumikhin-type condition (iii),
Applying Itô formula and by (3.16), one obtains that
Finally, by (3.9), (3.13), (3.14), and the Gronwall inequality,
which is a contradiction. So inequality (3.12) holds and (3.11) is true for .

Now assume that
for all , where . We proceed to show that
Suppose (3.20) is not true, set . By condition (ii) and (3.20), we know
From this, together with being continuous on , we know that and
Define , then and
For and , when , then (3.22) and (3.23) imply that
If for some , we assume that, without loss of generality, for some , then from (3.19) and (3.23),
Therefore,
Then, it follows from condition (iii) that
Combining Itô formula with (3.27), we can check that
Finally, by (3.22), (3.23), and the Gronwall inequality,
which is a contradiction. So inequality (3.20) holds. By mathematical induction, we obtain that (3.11) holds for all . Furthermore, from condition (i), we have
which implies
that is, system (2.1) is th moment exponentially stable. The proof is complete.

*Remark 3.3. *If , then Theorem 3.1 of [23] follows from Theorem 3.2 immediately.

Theorem 3.4. *Let , and let be a piecewise continuous function. Suppose there exist some positive constants , and such that** for all ,
** for all and ,
where with ,**for all , and ,
whenever
where .*

Then the trivial solution of system (2.1) is th moment exponentially stable and its th moment Lyapunov exponent is not greater than .

*Proof. *Given any initial data , the global solution of (2.1) is written as in this proof. Without loss of generality, assume that the initial date is nontrivial so that is not a trivial solution. Choose such that
Then it follows from condition (i) and (3.36) that
In the following, we will show that
where and is defined as and . Similarly, as the proof in Theorem 3.2, one can prove that
Now assume that
for all , where . We proceed to show that
Suppose (3.41) is not true, set . By condition (ii),
From this, together with being continuous on , we know that and
Define , then and
For and , when , then (3.44) implies that
If for some , we assume that, without loss of generality, for some , then from (3.41) and (3.44), we obtain
Therefore,
The rest of the proof is similar to that of Theorem 3.2 and omitted here.

*Remark 3.5. *Let and be positive constants. Assume that the conditions of Theorem 3.4 hold, function satisfies and . Then Theorem 3.1 of [24] follows immediately.

*Remark 3.6. *It is not strictly required by condition (ii) of Theorem 3.4 that each impulse contributes to stabilize the system, as long as the overall contribution of the impulses are stabilizing. Without these (i.e., ), it is required that each impulse is a stabilizing factor (), which is more restrictive.

*Remark 3.7. *It is clear that Theorems 3.2 and 3.4 allow the continuous dynamics of system (2.1) to be unstable, since the function , which characterizes the changing rate of at , is assumed to be nonnegative. Theorems 3.2 and 3.4 show that an unstable stochastic delay system can be successfully stabilized by impulses.

The following theorems show that the trivial solutions of system (2.1) are also almost surely exponentially stable, under some additional conditions.

*Assumption 3.8. *Suppose the impulsive instances satisfy

*Assumption 3.9. *Assume that there is a constant such that, for all ,

Lemma 3.10 (see [23]). *Let , and let Assumptions 3.8 and 3.9 hold. Then (3.31) implies that, for all ,
**
where is a positive constant. In other words, under Assumptions 3.8 and 3.9, the th moment exponential stability implies the almost exponential stability for system (2.1).*

By using Theorems 3.2 and 3.4 and Lemma 3.10, it is easy to show the following conclusions.

Theorem 3.11. *Suppose that , Assumptions 3.8 and 3.9 and the same conditions as in Theorem 3.2 hold. Then the trivial solution of system (2.1) is also almost surely exponentially stable, with its Lyapunov exponent not greater than .*

Theorem 3.12. *Suppose that , Assumptions 3.8 and 3.9 and the same conditions as in Theorem 3.4 hold. Then the trivial solution of system (2.1) is also almost surely exponentially stable, with its Lyapunov exponent not greater than .*

#### 4. An Example

*Example 4.1. *Consider a scalar ISDDs of the form
It is easy to check that the corresponding system without impulses is not mean square exponentially stable. In fact, if , then it follows from the Itô formula that . This leads to for all . But, in the following, we will show that system (4.1) is mean square exponentially stable and almost exponentially stable.

If , then condition (i) of Theorem 3.2 holds with , and condition (ii) holds with . By calculating, we have . By taking , and , it is easy to verify that condition (iii) of Theorem 3.2 is satisfied, which means system (4.1) is mean square exponentially stable. Applying Theorem 3.11, we can derive that system (4.1) is almost exponentially stable.

#### Acknowledgments

The authors are grateful to Editor Professor Josef Diblík and anonymous referees for their helpful comments and suggestions which have improved the quality of this paper. This work is supported by Natural Science Foundation of China (no. 10771001), Research Fund for Doctor Station of Ministry of Education of China (no. 20113401110001, no. 20103401120002), TIAN YUAN Series of Natural Science Foundation of China (no. 11126177), Key Natural Science Foundation (no. KJ2009A49), Talent Foundation (no. 05025104) of Anhui Province Education Department, 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 090416237, no. 1208085QA15), and Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).

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#### Copyright

Copyright © 2012 Zheng Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.