Solvability And Control For Some Evolution Systems Of Fractional Order

In this thesis,we mainly focus upon the solvability and control of some Sobolev type fractional equations of order(1,2)through some analytical properties of a class of resolvent operator family,and fundamental theories and techniques of fractional calculus and functional analysis.The main contents are concerned with the solvability and optimal control of Sobolev type stochastic fractional evolution system,the solvability and optimal control of Sobolev type fractional hemivariational inequalities,and the approximate controllability of Sobolev type stochastic fractional hemivariational inequalities.The obtained results improve and generalize some known ones.The thesis consists of six chapters.The first chapter is mainly involved in the background,advance of concerned topic,and the sketch of this thesis.The second chapter is concerned with some preliminary results such as some spaces of functions,basic results on fractional calculus,resolvent operator family,multivalued analysi and some theorems and lemmas needed in this thesis.In the third chapter,we mainly discuss the solvability and optimal control of Sobolev type stochastic evolution equations of order(1,2).We firstly establish the existence of mild solutions to the addressed system under mixed Carath′eodory and Lipschitz conditions,And then we present the existence of optimal control pairs of the corresonding limited Lagrange optimal systems.In the fourth chapter,we mainly investigate the solvability and optimal control of Sobolev type fractional evolution hemivariational inequalities of order(1,2).the existence of the hemivariational inequalities system is established.And then we present the existence of optimal state-control pairs for the limited Lagrange optimal systems governed by hemivariational system via fractional resolvent operators and the generalized Clarke subdifferential.In the fifth chapter,we mainly consider approximate controllability of Sobolev type stochastic fractional hemivariational inequalities of order(1,2).Sufficient conditions are established for the existence and approximate controllability of addressed systems.Finally,in the sixth chapter we give summaries and other problems to be further investigated in the future.