Random Pulse Exponential Stability Of Functional Differential Equations

The most classical and principle approach of investigating the stability of stochastic differential systems is Lyapunov's direct methods, which introduces an assistant function called Lyapunov function and uses its properties to judge whether the systems is stable or not. In this paper, by using the Lyapunov's direct methods, coupled with Razumikhin technique and comparison principle, we study exponential stability of the trivial solution of stochastic functional differential systems with random impulsive effect.This paper contains five chapters.Chapter 1 is the preface. We mainly introduce the developing history of impulsive functional differential systems, some achievements for stability made in this field and the significance of this study. At the same time, we present the organizing structure of this paper and some necessary preliminary notes.In chapter2, we discuss all kinds of stability of stochastic functional differential equations, which is prepared for the study of stability of the stochastic impulsive functional differential equations in the future.In chapter3, we discuss the p-moment stability of the trivial solution of delay differential equations with impulsive effect on fixed moments by means of Lyapunov's direct method.In chapter4, by means of Lyapunov's direct method coupled with Razumikhin technique and comparison principle, we study exponential a-nd asymptotic exponential stability of the zero solution of the stochastic differential equations when it is affected by impulsive on random moments and the quantity of impulsive is also random.In chapter5, we extend the model to the stochastic impulsive functional differential equations and continue to discuss exponential stability of the zero solution of the stochastic functional differential equations with random impulsive effect by means of Lyapunov's direct method coupled with Razumikhin technique and comparison principle.