On Some Problems For Stochastic Delay Evolution Systems

Stochastic delay evolution systems is an important topic of dynamics research. Based on the existed literatures, taking the effects imposed by stochastic perturbations, impulsive perturbations and delays into consideration, this thesis intensively studies four important problems of stochastic evolution systems:existence and uniqueness, controllability, approx-imately controllability and stability. According to the research of the problems, we discuss these important dynamic properties of the system.In Chapter 1, we introduce the historical background and the current situation of the problems to be studied. The main results of this thesis are also briefly demonstrated.In Chapter 2, some existence results for stochastic delay evolution equations are ob-tained. First, under the monotonicity, coercivity and local Lipschitze conditions, we get the local existence and uniqueness results by variational method and Galerkin approximation technique. Then, a continuation theorem for stochastic delay evolution systems is given with the help of stochastic analysis techniques and quasi-bounded condition. Some known results are generalized.In Chapter 3, by establishing a stochastic inequality on infinite delay, some approxi-mate controllability results for stochastic partial differential systems with infinite delays are obtained with the aid of fractional semigroup theory and stochastic analysis techniques. The cases that the control function nonlinearly affects the system and the delay is unbounded are also taken into consideration.In Chapter 4, we study existence of solutions for impulsive stochastic functional differ-ential inclusions. In the case where the right hand side is convex or nonconvex valued, we use Dhage fixed point theorem and Banach contraction principle for multi-valued operators and obtain the existence of the mild solutions. We also pointed out that the controllability problems could be studied with the similar approach.In Chapter 5, We investigate the mean square stability of impulsive stochastic func-tional differential equations with infinite delay. Some new criteria are obtained by using Lyapunov function method and stochastic analysis technique. We also give some examples to demonstrate our results.In Chapter 6, we discuss dynamic behaviors of stochastic neural networks with distributed delays. A condition to ensure the existence, uniqueness and uniformly LP bounded property of the solution is obtained by using Banach contraction principle. And then, a new criteria on pth exponential stability is obtained via stochastic analysis and differential inequality techniques.