Solvability And Controls For Some Multi-term Fractional Evolution Equations

This paper is mainly concerned with the solvability and controls of several kinds of multi-term fractional evolution equations.Combined with the properties of resolvent operator family,optimal control theory,fixed point technique et al.,firstly,the existence of solutions is investigated for a multi-term Caputo fractional evolution equation,and then the control problems of related system are explored.Secondly,based upon the investigation of the aforementioned semi-linear equation,the Sobolev type(or degenerate type)fractional evolution equations are also considered.On the one hand,we discuss the existence,generation theorem,asymptotic decay of resolvent operator family related to Sobolev type fractional evolution equations and time optimal controls of the associated systems;On the other hand,we consider the asymptotic behavior of solutions of Sobolev type fractional evolution equations and S-asymptotically Bloch type periodic solutions of Weyl type fractional evolution equations.The whole thesis is divided into six chapters:The first chapter is mainly involved in some backgrounds and main outline of the thesis.In Chapter 2,we introduce some basic results notions,and notations which are needed throughout the thesis.In Chapter 3,by the properties of resolvent operator family and fixed point technique,we deal with the solvability and optimal controls of a semi-linear multi-term fractional evolution equation with nonlocal initial conditions.In Chapter 4,we first define resolvent operator family related to Sobolev type evolution equations,and then investigate the time optimal control of Sobolev type multi-term fractional evolution equations by the properties of resolvent operator family and fixed point technique.In Chapter 5,Combined with the properties of resolvent operator family and fixed point technique,we study the asymptotic behavior of solutions of Sobolev type Caputo and RiemannLiouville fractional evolution equations and S-asymptotically Bloch type periodic solutions of Weyl fractional evolution equations respectively.In Chapter 6,the summary and prospect of this paper are given briefly.