Research Article | Open Access

Nan Ding, "Exponential Stability in Mean Square for Neutral Stochastic Partial Functional Differential Equations with Impulses", *Journal of Applied Mathematics*, vol. 2013, Article ID 729159, 8 pages, 2013. https://doi.org/10.1155/2013/729159

# Exponential Stability in Mean Square for Neutral Stochastic Partial Functional Differential Equations with Impulses

**Academic Editor:**Vu Phat

#### Abstract

We discuss the exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses. By applying impulsive Gronwall-Bellman inequality, the stochastic analytic techniques, the fractional power of operator, and semigroup theory, we obtain some completely new sufficient conditions ensuring the exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses. Finally, an example is provided to illustrate the obtained theory.

#### 1. Introduction

Stochastic partial differential equations have attracted the attention of many authors, and many valuable results on the existence, uniqueness, and stability of mild solution have been established. For example, Ren and Sakthivel [1] have established the existence and uniqueness of mild solution for a class of second-order neutral stochastic evolution equations with infinite delay and Poisson jumps by means of the successive approximation and the continuous dependence of solutions on the initial data by means of a corollary of the Bihari inequality; Sakthivel et al. [2, 3] have discussed the existence and uniqueness of square-mean pseudo almost automorphic mild solutions for stochastic fractional differential equations by using the stochastic analysis theory, fixed point strategy, and the existence of mild solution to nonlinear stochastic fractional differential by using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directed from deterministic fractional equations, respectively; Sakthivel et al. [4, 5] also have derived the exponential stability of mild solutions to the second-order stochastic evolution equations with Poisson jumps by applying stochastic analysis theory, and the existence and asymptotic stability in th moment of mild solution to second-order nonlinear neural stochastic differential equations with the help of fixed point theory, stochastic analysis technique, and semigroup theory, respectively. Besides stochastic effects, impulsive effects likewise exist in real-world models. It is to be noted that there has been increasing interest in the study of the existence, uniqueness, and stability of mild solution for stochastic partial functional differential equations with impulses due to its wide applications in various sciences, and many significant results have been obtained [6–15].

To the best of the author’s knowledge, there are only few works about the exponential stability of mild solution to neutral stochastic partial functional differential equations with impulses. One of the reasons is that the mild solutions do not have stochastic differentials, so Itô formula fails to deal with this problem. Another reason is that when we consider the exponential stability of mild solution for stochastic partial functional differential equations with impulses, impulsive effects on the system brings about many difficulties, since the corresponding theory for such problem has not yet been fully developed. For example, Sakthivel and Luo [8, 9] have discussed the asymptotic stability for mild solution of impulsive stochastic partial differential equations by using the fixed point theorem which can be regarded as an excellent tool to derive the exponential stability for mild solution to stochastic partial differential with delays in Luo [10, 11]; this very useful method may be difficult and even ineffective for the exponential stability of such system with impulses; some other methods used in Caraballo and Liu [12], Wan and Duan [13], and so forth are also ineffective for this problem, since mild solutions do not have stochastic differentials; in addition, Chen [14] established an impulsive-integral inequality to investigate the exponential stability of stochastic partial differential equation with delays and impulses, but it is not effective for neutral type. For the previous reasons, Long et al. [15] established a new impulsive-integral inequality to investigate the global attracting set and exponential -stability of stochastic neutral partial functional differential equations with impulses. However, the methods of studying the exponential stability of mild solution for neutral stochastic partial functional differential equations with impulses is still not abundant.

Motivated by the previous discussion, in this paper, we discuss exponential stability in mean square of mild solution for neutral stochastic partial functional differential equations with impulses. By applying impulsive Gronwall-Bellman inequality, the stochastic analytic techniques, inequality technique, the fractional power of operator, and semigroup theory, we obtain some completely new sufficient conditions to ensure the exponential stability in mean square of mild solution for stochastic partial functional differential equations with impulses.

The rest of this paper is organized as follows. In Section 2, we present some basic notations, definitions, and auxiliary results. In Section 3, sufficient conditions are derived to ensure the exponential stability in mean square for mild solution. Finally, an example is given to demonstrate the obtained results.

#### 2. Preliminaries

Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets).

Let be real separable Hilbert spaces, let and be the space of bounded linear operators mapping into . For convenience, we will use the same notations to denote the norms in , and without any confusion. Let denotes a -valued -Wiener process defined on with covariance operator ; that is, where is a positive self-adjoint, trace class operator on , denotes the inner product of , and denotes the mathematical expectation. In particular, we call such , a -valued -Wiener process with respect to .

In order to define stochastic integrals with respect to the -Wiener process , we introduce the subspace of which, endowed with the inner product , is a Hilbert space. We assume that there exists a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , and a sequence of independent Brownian motions such that and , where is the sigma algebra generated by . Let denote the space of all Hilbert-Schmidt operators from into . It turns out to be a separable Hilbert space, equipped with the norm Clearly, for any bounded operators , this norm reduces to .

Let and be the sets of real and integer numbers, respectively; let and denote the space of continuous mapping from the topological space to the topological space . Particularly, denotes the family of all continuous -valued functions defined on with the norm , where is a positive constant.

is continuous for all, but at most a finite number of points and at these points , and exist, , where is a bounded interval and and denote the right-hand and left-hand limits of the function , respectively. Particularly, let .

Let denote the family of all bounded -measurable, -valued random variables , satisfying .

In this paper, we consider the following neutral stochastic partial functional differential equation with impulses: where is the infinitesimal generator of an analytic semigroup of linear operator on a Hilbert space ; and are jointly continuous functions; the fixed moment of time satisfies , and ; and represent the right and left limits of at , respectively; represents the jump in the state at time with determining the size of the jump.

We also assume that , the resolvent set of . Then, it is possible to define the fractional power for some as a closed linear operator with its domain ; furthermore, the subspace is dense in , and the expression defines a norm on .

Lemma 1 (see Pazy [16]). *Suppose that , then, we know that there exist constants such that for , and for every *(i)* we have for each ,
*(ii)* there exists such that
*

*Definition 2. *A stochastic process , is called a mild solution of the system (4), if(i) is an adapted process,(ii) has a càdlàg path on almost surely,(iii) for arbitrary , we have
where , a.s.

*Definition 3. *The mild solution of system (4) is said to be exponentially stable in mean square if there exists a pair of positive constants and such that for any solution with the initial condition ,

#### 3. Exponential Stability

For system (4), we impose the following assumptions.(A1) There exist constants such that for any and , (A2) There exist such that for any and , (A3) There exist some positive numbers such that

for any and .

Under the assumptions, (A1)–(A3), the existence and uniqueness of mild solution to the system (4) are easily shown by using Picard iterative method.

Theorem 4. *Suppose that the assumptions (A1)–(A3) hold; furthermore, and the following assumptions*(A4)*,*(A5)*hold for , where is the Gamma function and , and are corresponding constants in Lemma 1. Then the mild solution of system (4) is exponentially stable in mean square.*

*Proof. *From (8), for any , we can get
It follows from (A2) that
By (A2), we can get
Employing Lemma 1, (A2), and Hölder inequality, we obtain
Combining Lemma 1 and (A3), with Hölder inequality, we can get
From Lemma 1, (A1) and Hölder inequality, we obtain
Using (A1) and Burkholder-type inequality, we obtain

Substituting (14)–(19) into (13), we haveFrom assumption (A5), we can get
where , .

According to Gronwall-Bellmen’s inequality [17], we have
On the other hand, by (A4), one has
Thereby, (22) can be rewritten as
which implies the mild solution of system (4) is exponentially stable in mean square.

This completes the proof.

Theorem 5. *Suppose that all the conditions of Theorem 4 hold. Then the mild solution of system (4) is exponential stable almost surely.*

*Proof. *The proof is quite similar to the proof of Theorem 5.1 in [18], and we omit it here.

If the impulsive effects, , system (4) is turned into the following neutral stochastic partial functional differential equations:

Corollary 6. *Assume that (A1)-(A2) hold, and the following condition
**
holds for . Then the mild solution of system (25) is exponentially stable in mean square.*

If , then system (4) becomes stochastic partial functional differential equations with impulses as follows:

Corollary 7. *Assume that (A1) and (A3) hold, and the following conditions*(A4)*,*(A5)*hold. Then the mild solution of system (27) is exponentially stable in mean square.*

If and , then system (4) becomes stochastic partial functional differential equations as follows:

Corollary 8. *Assume that (A1) holds, and the following condition
**
holds. Then the mild solution of system (28) is exponentially stable in mean square.*

*Remark 9. *In [12], Caraballo and Liu have studied the exponential stability in -moment of mild solution to (28) by utilizing Gronwall inequality, and the monotone decreasing behaviors of the delays are imposed. Particularly, when , the condition ensuring the exponential stability in mean square is . From Corollary 8, it is obvious that our results improve the results in [12].

#### 4. Example

*Example 1*. We consider the following neutral stochastic partial functional differential equation with impulses:
where are constants and denote the standard cylindrical Wiener process.

Let . Define bounded linear operator by

Then we get where is the set of eigenvector of .

The bounded linear operator is given by with domain

We can easily know that is the infinitesimal generator of an analytic semigroup , in and Furthermore, we know that .

Let .

Then we can get From Theorem 4, we know that the mild of system (30) is exponentially stable in mean square provided that

#### 5. Conclusion

In this paper, neutral stochastic partial functional differential equation with impulses has been investigated. By utilizing impulsive inequality, the stochastic analytic technique, inequality technique, the fractional power of operator, and semigroup theory, some completely new sufficient conditions are obtained to guarantee the exponential stability of mild solution for neutral stochastic partial functional differential equation with impulses. Finally, it should be pointed out that it is possible to extend the results obtained in this paper to other neutral stochastic impulsive differential equations, such as neutral stochastic impulsive partial differential equations with Poisson jumps and so forth.

#### Acknowledgments

The author would like to thank the referee and the editor for their careful comments and valuable suggestions on this work. This work is supported by the Key Foundation of Chongqing Three Gorges University (no. 12ZD19).

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#### Copyright

Copyright © 2013 Nan Ding. 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.