Solvability And Control Problems Of Semilinear Fractional Differential Equations

Fractional differential equations has been widely used in optical systems,thermal systems,mechanical systems and other application fields.It has great theoretical significance and application values.This paper is devoted to the control problems of semilinear partial differential equations and inclusions involving the Riemann–Liouville fractional derivative with respect to time.That is,the solvability,optimal control and approximation controllability of semilinear fractional partial differential equations,and the solvability and approximation controllability of fractional differential inclusions.Firstly,based on the Banach fixed point theorem and semigroup theory,the existence,uniqueness and boundedness of the mild solutions to semilinear fractional differential equation are proved.Compared with the results in existing literature,our results is better since the assumption of nonlinear terms is improved.Based on the compactnesslemma in L~p(J;X),the Lagrange multiplier and nonlinear analysis techniques,and the optimal control theory of partial differential equations,the optimal control of semilinear fractional differential equations is considered.We show the existence of optimal solutions and derive their necessary optimality conditions of first order.Secondly,based on the study of fractional differential equations,the existence of mild solutions for Riemann-Liouville fractional differential inclusions is proved via using the fixed point theorem of set-valued mapping and the theory of nonlinear analysis and set value analysis.Finally,based on the previous work,we study the problems of approximation control-lability for fractional differential equations and inclusions,respectively.Their approximation controllability in the Banach space L~p(J;X)has been first proved.