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

Nan Ding

doi:10.1155/2013/729159

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.

Highlights

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
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
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], Journal of Applied Mathematics

Summary

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. Wan and Duan [13], and so forth are ineffective for this problem, since mild solutions do not have stochastic differentials; in addition, Chen [14] established an impulsiveintegral inequality to investigate the exponential stability of stochastic partial differential equation with delays and impulses, but it is not effective for neutral type. 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. An example is given to demonstrate the obtained results

Preliminaries

Exponential Stability

Example

Conclusion