Research On The Stability And Control Lability Of Several Kinds Of Fractional Evolution Equations

Within the past few decades,existence of solution(mild solution,weak solution),stability and controllability of fractional differential systems have been hot topics in the control field.This paper first deals with the existence of mild solution and controllability of two kinds of fractional evolution equations.Then the stability of solutions of the initial value problem for a class of time-fractional diffusion equations are studied.Finally,the existence and uniqueness of the weak solution of initial-boundary-value problems for a class of time-space fractional diffusion equations are investigated.The main contents of this thesis are arranged as follows.In the first chapter,the relative background and the significance of the research of fractional evolution equations are introduced briefly.Then the main work of the thesis are presented.The second chapter is concerned with some preliminary knowledge including some necessary notations and function spaces,the theory of fractional calculus,semigroup of operators and fixed point theorems which will be used in the paper.In chapter 3,the existence of mild solution and controllability of a class of fractional evolution equations are discussed.Firstly,based on the references[55,56],we give the definition of mild solution of the above system.Then we apply the theory fractional calculus,semigroup of operators,Schauder's fixed point theorem of the knowledge of nonlinear functional analysis to derive the existence of mild solution and approximate controllability of the system.Furthermore,the complete controllability of the system is investigated.At last,the approximate controllability of a fractional partial differential equation is discussed to explain the application of our results.The existence of mild solution and complete controllability of a class of fractional evolution equations are studied in chapter 4.Similarly,based on the references[55,56],we give the definition of mild solution of the equation.Then by using the theory of fractional calculus,semigroup of operators(fractional power and so on)and Banach's fixed point theorem,we derive the existence of mild solution and complete controllability of the equation.Finally,an example is given to show the application of the result.Chapter 5 is devoted to the stability of solutions of the initial value problem of a fractional diffusion equation.We transform the initial value problem into the correspond-ing homogeneous initial value problem(?)and inhomogeneous initial value problem(?)by the superposition principle.Then we apply fractional calculus,the Laplace transform,the Fourier transform to obtain the expression of solution of(?).Moreover,the fractional Duhamel principle for(?)is established.As a result,the solution of the initial value problem is given by the superposition principle.Also,the stability of solutions of the initial value problem of the fractional diffusion equation are obtained.At the end of this chapter,an example is presented to illustrate our results.The chapter 6 aims to study the existence and uniqueness of the weak solution of initial-boundary-value problems for a time-space fractional diffusion equation.We first establish the existence of the weak solution of the initial-boundary-value problem and the proof is based on the eigenfunction expansion.Then to prove the uniqueness of the weak solution,we present a maximum principle for the time-space fractional diffusion equation by using the properties of the time fractional derivative and the fractional Laplace operator.The results are extensions of some existed conclusions in[126].