Existence And Controllability Of Solutions For A Class Of Sobolev-type Fractional Evolution Equations

In this thesis,we mainly discuss the solvability and controllability of nonlocal problem of Sobolev-type fractional evolution equationWhere D_t~?is the Riemann-Liouville derivative of order ??(1,2),A and E are closed linear operators defined on a Banach space,u is the control function,B is a bounded linear operator.The main results of this thesis are as follows:1.Under conditions of the measure of non-compactness,we study the existence and exact controllability of mild solutions for the nonlocal problem(I)of Sobolevtype fractional evolution equation by using Sadovskii's fixed point theorem and the theory of resolvent operators.2.By utilizing Schauder's fixed point theorem and approximation technique,we investigate the existence and approximate controllability of mild solutions for the nonlocal problem(I)of Sobolev-type fractional evolution equation.3.By using the contraction mapping principle,we prove the existence and uniqueness of mild solutions for the nonlocal problem(I)of Sobolev-type fractional evolution equation.The optimal controls are also investigated.