Controllability And Stability Of Several Impulsive Stochastic Functional Differential Systems

This dissertation is mainly concerned with the existence and controllability for four kinds of infinite dimensional stochastic neutral functional differential systems with state-dependent delays in the Hilbert space frame,and the p th moment exponential stability for a class of impulsive stochastic functional differential systems with Marko-vian switching.The main work of this dissertation are as follows:The research background,significance and recent development,of finite dimensional stochastic differential equations and infinite dimensional stochastic differential systems are summarized in Chapter 1.Some preliminaries are briefly introduced in Chapter 2,including the theory of stochastic differential equations,Q-Wiener process and infinite dimensional stochas-tic.integral,poisson point process and poisson random measure,integro-different,ial(evolution)equations and resolvent operators theory,second order abstract differential equations theory,several fixed point theorems and inequalities.The existence and controllability for a class of first order impulsive neutral stochas-tic integro-differential equations with state-dependent delay are investigated in Chapter 3.Under the noncompactness assumption of resolvent operators,the sufficient condi-tions of the existence and controllability results are obt,ained by virtue of fixed point theorems,the theory of analytic resolvent operators,the theory of fractional power and a norm.At the end of this chapter,an example of the heat conduction equations with fading memory is illustrated to show the effectiveness of the main results.Chapter 4 is devoted to the existence and controllability for a class of first order im-pulsive neutral stochastic integro-differential evolution equations with state-dependent delay.Under some suitable conditions,the existence and controllability of mild so-lutions are obtained by virtue of Banach fixed point theorem,Sadovskii fixed point theorem and the theory of analytic resolvent operators.The results generalize some of the existing conclusions in the literature.Chapter 5 focus on the existence and controllability for a class of second-order im-pulsive neutral stochastic integro-differential evolution equations with state-dependent delay.First,under different assumptions and the theory of second order evolution e-quations,sufficient conditions of the existence of mild solutions for this equations are established by using Sadovskii fixed point theorem and Krasnoselskii-Schaefer fixed point theorem,respectively.Then,under some suitable conditions,the controllability results are obtained by employing the Banach fixed point theorem.Also,the result-s are applied to the second-order impulsive neutral stochastic wave equations with state-dependent delay,and we obtained some relevant conclusions.Chapter 6 is considered with the approximate controllability for a class of second-order impulsive neutral stochastic differential equations with state-dependent delay and poisson jumps.Under the appropriate assumptions,sufficient conditions of the approximate controllability results are obtained by virtue of the theory of a strongly continuous cosine family of bounded linear operators,Sadovskii fixed point theorem and stochastic analysis technique.Also,the results are applied to second-order impulsive neutral stochastic wave equations with state-dependent delay and poisson jumps,and obtained some relevant conclusions.In Chapter 7,some new stability criteria of p th moment exponential stability for first order impulsive stochastic functional differential equations with Markovian switch-ing are obtained using the Lyapunov function method and Razumikhin technique with stochastic version as well as stochastic analysis theory.Main results show that some unstable impulsive stochastic functional differential equations with Markovian switch-ing may be exponentially stabilized by impulses.Finally,two examples of numerical simulation are presented to illustrate the efficiency of main results.