Asymptotic Properties And Control Problems Of Solutions For Several Kinds Of Integro-Differential Evolution Systems

The theory of integro-differential evolution systems is an important branch of the theory of infinite dimensional evolution systems.Compared with the general differential equations,in many cases integro-differential evolution systems can more accurately describe the natural phenomena arising from many fields of science.Thus,the research of various dynamic behaviors for this kind of systems has vital theoretical and practical significance.In this dissertation,the effects of stochastic phenomena,impulsive phenomena,nonlocal conditions and time delay on the system are considered.By using theory of semigroups,resolvent operators,fractional powers of operators,fundamental solutions,stochastic analysis as well as fixed point theorems,we study some asymptotic properties,approximate controllability and optimal control problems of solutions for several kinds of integro-differential evolution systems.The work in this paper generalizes some existing conclusions in this field.The whole dissertation consists of five chapters.In Chapter 1,we first introduce some backgrounds and research status of integrodifferential evolution equations and present some recent relevant works on integrodifferential evolution equations.We then state briefly the main work of this dissertation.In Chapter 2,we study the asymptotic properties of solutions for a class of impulsive neutral stochastic functional integro-differential systems.By applying the theory of resolvent operators,Banach fixed point principle and results on stochastic analysis,we study respectively the existence,uniqueness,global attracting and quasi-invariant sets of mild solutions for the considered equation.Also,we derive some sufficient conditions on -th moment exponential stability and almost surely exponential stability of the mild solutions.In Chapter 3 we prove the approximate controllability of semi-linear neutral integrodifferential systems with nonlocal conditions.Since fractional power operators and -norm are used to discuss the problems,the obtain results in this chapter can apply to the systems involving partial derivatives of spatial variables.It is worth mentioning that the compactness condition or Lipschitz condition for the function 2)in the nonlocal condition appearing in literature is not required here.In Chapter 4,we first construct the fundamental solutions of linear integro-differential evolution systems,from which the expressions of mild solutions for a class of semi-linear stochastic integro-differential systems with infinite delay are obtained via the fundamental solution based on Laplace transform arguments.From this we show the approximate controllability of the considered systems through the so-called resolvent conditions.Due to the fundamental solution theory,the uniform boundedness for the nonlinear terms is partly overcame and the results in this chapter generalize the results in literature.By virtue of the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature,Chapter 5 first discusses the existence and uniqueness of mild solutions and shows the compactness of solution operators for the neutral integro-differential system with infinite delay.Then optimal control and time optimal control problems for the considered control system are investigated under some assumptions.