Solvability And Control Of Impulsive Stochastic Functional Integro-differential Systems

Impulsive stochastic functional integro-differential system is one of important branch in the theory of nonlinear analysis,and it contains the effects imposed by stochastic phenomena,impulsive phenomena and delay,which has wide applications in many fields,such as engineering,economics,optimal control,information and com-munication,biology and medicine and so on.Therefore,it has vital theoretical and practical significance to study the solvability,controllability,approximate controlla-bility and optimal control for such systems.The present dissertation focuses on the impulsive stochastic functional integro-differential equations and integro-differential inclusions with not instantaneous im-pulses in Hilbert spaces.By using theory of resolvent operators,fractional power of closed operator,theory of stochastic analysis and measure of noncompactness,we firstly discuss the existence of mild solutions to several types of impulsive stochastic functional integro-differential systems with not instantaneous impulses.Then the re-sults are extended to the related control problems described by these systems.The present dissertation consists of five chapters.In Chapter 1,we introduce briefly the background,our main work and some preliminary knowledge required for this article.In Chapter 2,by means of the Hausdorff measure of noncompactness,the theory of analytic resolvent operators and fractional power of closed operator with Darbo fixed point theorem and Darbo-Sadovskii fixed point theorem,we consider the ex-istence of mild solution to a class of first order impulsive stochastic partial neutral functional integral-differential equations with infinite delay and not instantaneous im-pulses under the condition of non-compactness,and obtain some new results.In Chapter 3,we discuss the solvability and controllability of a class of first order impulsive stochastic partial neutral functional integral-differential inclusions with state dependent delay and not instantaneous impulses.Through introducing a suitable?-norm function space,we establish the existence of?-mild solutions and extreme?-solutions for the systems by using stochastic analysis,the theory of analytic resolvent operators,fractional power of closed operator and Dhage fixed point theorem of multi-valued mapping.On this basis,the controllability of the stochastic control systems with not instantaneous impulses is obtained.In Chapter 4,by using the H¨older inequality,the analytic?-resolvent operator,stochastic analysis,fractional calculus,fractional power of closed operator and Dhage fixed point theorem of multi-valued mapping,we study the approximate controllability of a class of fractional order impulsive stochastic partial neutral functional integral-differential inclusions with infinite delay and not instantaneous impulses under the Lipschitz and Carath?eodory conditions.The result is based on the corresponding linear integral-differential system is controllable.In Chapter 5,through introducing a suitable phase space B_h,by using the H¨older inequality,stochastic analysis,the theory of analytic semigroup,linear evolution sys-tem and fractional power of closed operator with the Krasnoselskii-Schaefer type fixed point theorem,we derive the optimal control of a class of first order impulsive stochas-tic neutral evolution integral-differential equations with infinite delay and not instan-taneous impulses in an?-norm function space.