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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 707069, 8 pages
http://dx.doi.org/10.1155/2015/707069
Research Article

Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance

1Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
2Control and Simulation Center, Harbin Institute of Technology, Harbin 150001, China

Received 27 February 2014; Accepted 5 August 2014

Academic Editor: Weihai Zhang

Copyright © 2015 Liming Wang 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.

Abstract

This paper considers the -moment boundedness of nonlinear impulsive stochastic delay differential systems (ISDDSs). Using the Lyapunov-Razumikhin method and stochastic analysis techniques, we obtain sufficient conditions which guarantee the -moment boundedness of ISDDSs. Two cases are considered, one is that the stochastic delay differential system (SDDS) may not be bounded, and how an impulsive strategy should be taken to make the SDDS be bounded. The other is that the SDDS is bounded, and an impulsive disturbance appears in this SDDS, then what restrictions on the impulsive disturbance should be adopted to maintain the boundedness of the SDDS. Our results provide sufficient criteria for these two cases. At last, two examples are given to illustrate the correctness of our results.

1. Introduction

Boundedness is an important property of a given system; for example, in the population models, the boundedness of a biological population is strongly connected with the persistence and extinction [1]. Another important application is on the stability; the practical stability actually is of a kind of boundedness [2]. Impulsive phenomena widely exist in the real world, and known, impulsive effects can change the properties of a given system; for example, given an unstable system, if a suitable impulsive strategy, including the impulsive strength and impulsive moments, is adopted, this system can be stabilized [3]. It is easy to understand that the impulsive effects can destroy the boundedness of a given system when the impulsive strength is large enough and the impulsive interval is small enough. Time delay is extensive in the engineering and applications and impulsive delay differential systems were considered in lots of papers [39]. The boundedness of impulsive delay differential systems has also been paid considerable attentions in the past decades. In [10], the authors presented sufficient conditions for uniform ultimate boundedness by virtue of the Lyapunov functional method. The boundedness of variable impulsive perturbations system was considered in [11] and the eventual boundedness was studied in [12]. Recently, the perturbing Lyapunov function method was also used in the study of boundedness [13].

Stochastic noise is ubiquitous [1416] and stochastic delay differential systems (SDDSs) have been one of the focuses of scientific research for many years. Many properties of SDDSs have been studied and lots of papers were published; see [17, 18] and the references therein. Being the wide existence of stochastic delay and impulsive effects, it is a natural task to consider the stochastic delay differential systems with impulsive effects. These systems are described by impulsive stochastic delay differential systems (ISDDSs). In the past ten years, the stability of ISDDSs has attracted a lot of researchers, and a great deal of results on the stability of ISDDSs have been reported; see [1924] and the references therein.

However, little attention has been paid to the boundedness of ISDDSs. In this paper, the boundedness of ISDDSs is considered under two cases. The first case is that the SDDSs may be unbounded, then what kind of impulsive strategy should be taken to make the system be bounded. The second case is that the SDDSs are bounded, then this system can tolerate what kind of impulsive effect to maintain the boundedness.

In this paper, sufficient conditions are presented to guarantee the boundedness of ISDDSs; these conditions also admit the global existence of solutions for ISDDSs, which usually was a standard assumption in many papers [2527]. Making use of the Lyapunov-Razumikhin method, we generalize the results of [10] to the stochastic situation. At last, two examples are given to illustrate the correctness of our results.

2. Preliminaries and Model Description

Let be a complete probability space with a filtration satisfying the usual conditions (i.e., the filtration contains all -null sets and is right continuous). Let , , and . If is a vector or a matrix, its transpose is denoted by . Consider , is continuous for all but at most countable points and at these points, and exist and , where is an interval and and denote the right-hand and left-hand limits of the function at time , respectively. Consider and if is not at the uncontinuous points . Let denote the family of all bounded measurable, valued random variables. Let be the Euclidean norm in and .

Consider the following nonlinear impulsive stochastic delay differential system:where , , , and satisfies global Lipschitz condition, represents the delay in system (1), impulsive moment satisfies , and as . is an dimensenal Brownian motion and .

Given a function , the operator of with respect to system (1) is defined bywhere

Definition 1. System (1) is said to be (1)moment bounded if, for every and , there exists such that if with and is a solution of (1), then for all ;(2)moment uniformly bounded if the system (1) is moment bounded and is independent of ;(3)moment ultimately bounded if the system (1) is moment bounded and there exists a positive constant such that for every and there exists some ; if with , then for ;(4)moment uniformly ultimately bounded, if the system (1) is moment ultimately bounded and is independent of .

3. Boundedness with Impulsive Control

In this section, we consider the first case: when the given SDDS may not be bounded, we adopt an impulsive strategy to get the boundedness. The main result is stated as follows.

Theorem 2. Assume there exist a positive function and positive constants , where and , such that (1) for any ;(2)for , any , and , whenever and ;(3) for all ;(4)there exists a positive constant such that if , then ;(5), .
Then the system (1) is moment uniformly ultimately bounded.

Proof. We separate the proof into two parts. First, we show the moment uniform boundedness and then we give the ultimate uniform boundedness.
Step 1. Let . Without loss of generality, we assume . Choose such that ; then we can see .
Let and for some positive integer . Suppose is a solution of system (1) with initial value and its maximal interval of existence is for some positive constant . We will show that, for any , . By the way, if this statement is true, we know that the solution of system (1) is not explored in , and the global existence of the solution follows.
For the sake of contradiction, suppose for some . Then there exists . Note that for ; we see that and for and .
Write . For , we have , and . Define and then and for and .
We claim that for any and then .
If it is not true, suppose for some . If , then , which is a contradiction. If , then . Then , which is a contradiction.
Now we will proceed under two cases.
Case 1. Consider .
Let . Since , , and is continuous on , then and and, when , . Hence, for and , we haveand we can getThen, by virtue of condition (2), for ,However,which is contradiction. Then we get, in this case,Case 2. Consider for some .
Note that . This inequality can be obtained by the following reason: if , then . If , we get , and thenDefine , and then , , and for . The same argument as the one in Case 1 yields a contradiction. Therefore, in this case, we have, for any ,Now we get that, under conditions (1) to condition (5), the solutions of (1) are moment uniformly bounded. That is, if , there exists a constant , such that for all , and, from the proof, we have .
Step 2. Now, let and assume, without loss of generality, that . Then, from the proof of uniform boundedness, there exists some for which if , then for .
Take a constant satisfying ; it is easy to verify that . Let be the smallest positive integer for which and . Given a solution where and , we will show for .
Given a constant satisfying and , we will show that if for , then for .
For the sake of contradiction, suppose that there exists some for which and defineand we suppose for some . We can get for and .
We claim that . The fact follows that if , then . If and we have , then .
Now, since , we have and . This implies that ; that is, and since is continuous at . Also, for , we have . DefineSince , we have and and for . Then, if and ,which yields . Then, in light of condition (2),In terms of Itô formula,Butand this contradiction proves that for all .

Now we define a sequence , satisfying and , and then we have . By induction, we get . We know that when , that is, , ; then by induction we get for and then for . Using condition (1), we get that ; that is,

Remark 3. Condition (2) means the system without impulse may be unbounded. If the impulsive effects satisfy condition (3) to condition (5), then this system can be bounded.

4. Boundedness with Impulsive Disturbance

In this section, we consider the case that the SDDS is bounded, and when the impulsive disturbance appears in the SDDS, then what restrictions should be added to the disturbance to maintain the boundedness. The result is stated as follows.

Theorem 4. Assume that there exist a positive function and positive constants , where , such that (1) for any ;(2)for , any , and , whenever and ;(3) for all ;(4)there exists a positive constant such that if , then ;(5)there exist positive constants and , such that and .
Then, the system (1) is moment uniformly ultimately bounded.

Proof. Step 1. Let ; without loss of generality, we assume . Choose , such that , and then we get . Let and assume ; moreover, we assume that (1) has a maximal interval of existence, .
We will prove that for . This will show that and that solutions of (1) are uniformly bounded.
For the sake of contradiction, we suppose that for some . Let . Note that for , and we get , for and .
For , we have and then . Particularly, and .
Define and then , , and for .
Now we will proceed under two cases.
Case 1. Consider .
Under this case, we have   because of the continuity of on and . Define and then , , and for . Therefor, for any and , we have and , which yields , and then we have . Using condition (2), we have, when ,By virtue of Itô formula, we haveHowever,This contradiction givesCase 2. Consider for some .
We first show . We have two situations to contemplate: and .
If , we suppose . Define and then and . In light of the definition of , we have, for and ,and, for ,By virtue of condition (2), an analogous calculation of yields ; then we getIf , we suppose . We will proceed under two subcases.
Subcase 1. Consider for all .
Under this situation, we have and for all and . In terms of condition (2), an analogous discussion as done in Case 1 givesHowever, by virtue of condition (5),This contradiction impliesSubcase 2. Consider for some .
Define and then and . Using the definition of , we get, for and , . Since , using the fact and , we can get . By virtue of condition (2), we get, for , An analogous discussion as done in the case gives . Then we haveThis contradiction givesNow we claim . If , we get . If , we get . That is, the following inequality holds:Since , we have and .
If for all , then let and we have . Otherwise, let , and we have . Since , we get . Moreover, for , we have and, by virtue of , we obtain and then . In terms of condition (2) and Itô formula, we can obtain . But , which is a contradiction and yieldsNow we get that, under condition (1) to condition (5), the solutions of (1) are moment uniformly bounded. Then we know that if , there exists a constant , such that for all , and, from the above proof, we have .
Step 2. Now, let and assume, without loss of generality, that . Then, from the proof of uniform boundedness, there exists a constant for which if , then for .
Take a constant satisfying , , and .
Let be a solution of (1) with , . We will show for .
Given a positive number satisfying and , we will show that if for and , then for and .
For the sake of contradiction, suppose that there exists a constant for which and defineNote that , and we have that if , then . If , we have . Then we get , , and for .
If for all , we let , and then . Otherwise, let , and we get . Since , . For and , we have . Moreover, for ,and we get . By virtue of condition (2) and Itô formula, we can get . However, .
Now we have proven for , and we are on the position to show . This will follow in the same way as the arguments used in the proof of uniform boundedness, where we show for the case ; we just need to replace by and by .
By induction, we get that if for and , then for all and for .
Next, we will show for , if for all and , .
We first show . This can be easily verified under two situations: iIf , we have ; if , .
In order to verify for all , suppose that for some . Let ; we know and then and .
If for all , let , .
If for some , let and we know , .
For and , and , and we get . In terms of condition (2) and Itô formula, we can get . However, , which yieldsApplying our results to successive intervals of the form for , we can get for .
Now we need a fact . This can be verified just as we did in the proof of uniform boundedness, where we show for the case .
Take satisfying . Take , when , and we get . Since , we have when . By virtue of condition (1), for , which completes the proof.

Remark 5. Theorem 4 considers that a bounded system without impulse can tolerate what kind of impulsive effects to hold the boundedness. It is not surprising that condition (3) to condition (5) should be satisfied: the interval of impulsive moments () should be large and impulsive strength () should be small.

5. Examples

In this section, we present two examples to illustrate our results.

Example 1. Consider the following impulsive stochastic delay differential system:where is a one-dimension Brownian motion.
Define ; the smoothness requirement is satisfied. Let and ; condition (1) of Theorem 2 follows. For any solution of system (36), we haveTake ; condition (3) of Theorem 2 is satisfied.
Now let ; then, when and , that is, , we have Then let ; condition (2) of Theorem 2 is verified.
Condition (4) of Theorem 2 can be verified by taking .
Take and then ; condition (5) of Theorem 2 is verified.
Therefore, according to Theorem 2, solutions of system (36) are mean square uniformly ultimately bounded. The boundedness can be read from Figure 1, where we take initial condition .
To see the contribution of impulsive effect on boundedness, we consider the following system:which is the situation of system (36) without impulses. It is easy to be verified that system (39) is unbounded; see Figure 2, where we also take initial condition .

Figure 1: Mean square uniform ultimate boundedness of solution of system (36).
Figure 2: Unboundedness of solution of system (39).

Now we give another example to illustrate the correctness of Theorem 4.

Example 2. Considerwhere is a one-dimension Brownian motion.
Define ; the smoothness requirement is satisfied. Let and ; condition (1) of Theorem 4 follows. For any solution of system (40), we haveTake , condition (3) of Theorem 4 is satisfied.
Now let and ; then, when and , that is, , we haveThen, let ; condition (2) of Theorem 2 is verified.
Condition (4) of Theorem 2 can be verified by taking .
Take and then and condition (5) of Theorem 4 is verified.
Therefore, according to Theorem 4, solutions of system (40) are mean square uniformly ultimately bounded. The boundedness can be seen in Figure 3, where we take initial condition .
We also present the simulation of system (40) without impulsive effects; that is,The property of system (43) can be read from Figure 4, where we take initial condition .

Figure 3: Mean square uniform ultimate boundedness of solution of system (40).
Figure 4: Simulation of system (43).

Conflict of Interests

The authors declare that there is no conflict of interests regarding to the publications of this paper.

Acknowledgments

This work is supported by the foundation under Grant HIT.IBRSEM.A.2014015 and by the National Natural Scientific Foundation of China under Grants 11271101, 61104193, and 61333001.

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