Approximate Controllability Of Two Kinds Of Semi-linear Integro-differential Evolution Systems

As a kind of important evolution systems,integro-differential systems have a wide application background.The study of control problems of integro-differential evolution systems has great value both in theory and in practice.In this thesis,we mainly study the problem of approximate controllability for two kinds of semilinear integro-differential systems by using the theory of analytic semigroup,resolvent operators,theory of fractional power operators and fundamental solutions.The obtained results extends the existing work in related literature.The whole thesis contains three chapters.In Chapter 1,we present the research background,and state briefly the main work of this dissertation.In Chapter 2,we study the approximate controllability of a class of semilinear integrodifferential systems with finite delay.We first construct the theory of fundamental solutions for the corresponding linear system to obtain the expression of the mild solutions of the semilinear integro-differential system with delay by Laplace transform.Then based on the existence and uniqueness of the solution,a sufficient condition of approximate controllability of the system is established by using techniques on fractional power operators?the theory of resolvent operators and the so-called resolvent condition.Finally,an example is given to illustrate the applications of the results.In Chapter 3,by constructing fundamental solutions and using the theory of Schauder fixed point and resolvent operators,we obtain sufficient conditions of approximate controllability for a semilinear integro-differential system with nonlocal condition in a Hilbert space.An example is also provided in the end.