The Study On Control Problems Of Fractional Evolution Equations And Inclusions

Fractional evolution equations and evolution inclusions are an important branch of nonlinear functional analysis,which are widely used in fractal geometry,biomathematics and viscoelastic systems.The research on their control has been the hot topic.Based on fractional calculus,semigroup theory,set-valued analysis and fixed point theorem,the controllability and optimal control for several kinds of fractional evolution equations and inclusions are studied in this thesis.The main contents are as follows:Firstly,the approximate controllability of a class of fractional evolution inclusions with damping is studied.In the absence of semigroup properties,based on the assumption of compactness and continuity in the uniform operator topological of the (α,μ)-regularized resolvent family,it is proved that the related operators generated by the the regularized operators also satisfy compactness and continuity in the uniform operator topological.Next,by using the nonlinear alternative of Leray-Schauder and Covitz-Nadler fixed point theorem,the existence of a mild solution for the system under Carathéodory condition and Lipschitz condition are proved,respectively.On this basis,two different sufficient conditions for approximate controllability of fractional evolution inclusions with damping are formulated and proved.Finally,the main results are illustrated with the help of two examples.Secondly,the approximate controllability of Hilfer fractional stochastic evolution inclusions with nonlocal conditions is discussed.A weighted continuous function space is introduced to solve the problem that the solution of Hilfer fractional evolution equations may have singular terms.Then the existence of the mild solution for the system is proved by using stochastic analysis theory,fractional calculus and fixed point theorem of set-valued mapping.Combined with the approximate controllability of the corresponding linear system,the suitable sufficient conditions for approximate controllability of the Hilfer fractional stochastic evolution inclusions with nonlocal conditions are obtained.Finally,an example is given to verify the rationality and validity of the theoretical results.Thirdly,the approximate controllability of a class of fractional impulsive evolution equations with infinite delay in separable reflexive Banach spaces is studied.Firstly,we give a proper definition of the resolvent operator in Banach space,and modify the characterization of phase space in the presence of impulse effect.Secondly,an optimal control problem is constructed for the corresponding linear control system,and the optimal control expression is derived based on the resolvent operator consisting of duality mapping.We assume only that the impulse function satisfies the continuity condition.The existence of the solution for the system is proved by using an approximating technique,fractional power operator and Krasnoselskii’s fixed point theorem.Then,based on the approximate controllability of the linear control system,several sufficient conditions for approximate controllability of the fractional neutral impulsive evolution system are obtained.At last,an example is given to verify the validity of the main results.Finally,the optimal control problem for a class of fractional nonautonomous evolution inclusions of Clarke’s subdifferential type is studied.Based on the noncompactness measure method,some properties of Clarke’s subdifferential,generalized Gronwall inequality and fixed point theorem of condensed set-valued mapping,the existence of the solution for the system is proved under some appropriate local growth condition and a noncompactness measure condition assumptions on the nonlinear function.Then,the existence of optimal control-state pairs for Lagrange type problems is discussed by using Marzur lemma,Balder theorem and control convergence theorem.Finally,we give a concrete application example to illustrate the main results.