Some Problems Of Fractional Evolution Systems

Fractional evolution equations have obtained comprehensive applications on the model-ing of many phenomena in various fields of engineering, physics and economics. There hasbeen a significant development in fractional evolution equations in recent years.On the other hand, diferential inclusions arise in the mathematical modelling of certainproblems in economics, optimal control, stochastic analysis, etc. and are widely investigatedby many authors.In this thesis, we shall be concerned with the existence of solutions for fractional evolu-tion systems, the approximate controllability and the optimal feedback control.The thesis is divided into seven chapters:In chapter1, we introduce the background, the domestic and foreign research situationand the main research of this thesis.In chapter2, we introduce the definitions and properties of fractional calculus, multival-ued map and measure of noncompactness, and some related lemmas which are used in thisthesis.In chapter3, by using the fractional power of operators and the theory of measure ofnoncompactness, we discuss a class of fractional neutral evolution equations with Riemann-Liouville fractional derivative. We establish sufcient conditions for the existence of mildsolutions for fractional neutral evolution equations in the cases C0semigroup is compactor noncompact. In the end, we give an example to illustrate the applications of the abstractresults.In chapter4, we establish sufcient conditions for the existence of solutions of frac-tional evolution inclusions involving the Riemann-Liouville fractional derivative. The casesof convex-valued and nonconvex-valued right-hand sides are considered and we present aversion of Filippov’s theorem for fractional semilinear diferential inclusions with Riemann-Liouville derivative.In chapter5, we investigate the approximate controllability problem for fractional func-tional evolution inclusions in Hilbert spaces. The results are obtained under the correspondinglinear system is approximately controllable. By using the fixed-point theorem for condensingmultivalued maps and semigroup theory, sufcient conditions are formulated and proved. Inthe end, we give an example to illustrate the applications of the abstract results. In chapter6, we consider the optimal feedback control problems of a system governed bynonlinear fractional impulsive evolution equations. Based on the existence of feasible pairs,an existence result of optimal control pairs for the Lagrange problem is presented. The resultswe obtained are a generalization and continuation of the recent results on this issue.In chapter7, we give a conclusion of our present research and introduce some furtherstudy ideas in the future.