The Research Of Impulsive Fractional Differential Equations And Control Systems

This thesis mainly focus on impulsive fractional differential equations and control systems. For impulsive fractional differential equations, some related existence, uniqueness and data dependence results are obtained. For impulsive fractional control systems, we discuss the controllability and observability. Ad-ditionally, we have an interesting conclusion about the necessary conditions of the optimal control for nonlinear fractional control system without impulsive. There are eight chapters in this thesis.Chapter1is introduction. The research background and the significance of this thesis are introduced, then our questions and main results are briefly presented.In Chapter2, the contents are about the fractional calculus. Also, some notations and preliminary facts are recalled.In Chapter3, a class of impulsive fractional functional differential equa-tions are investigated. Natural formula of a solution for the problem, which can be regarded as a novelty item, are introduced. Then we establish some new existence, uniqueness and data dependence results.In Chapter4, we discuss boundary value problem(BVP for short) for impulsive fractional differential equations. The first purpose is introducing a natural formula of solutions. The second purpose, by applying fixed point methods via a generalized singulary Gronwall inequality, is establishing some related new existence results.In Chapter5, the initial value problems (IVP for short) and nonlocal prob-lem for a kind of nonlinear impulsive fractional differential equations involving the Caputo fractional derivative in Banach space are considered, respectively. A suitable formula of the solutions is given. Some related existence results are obtained by utilizing the technique of the measures of non-compactness and Monch’s fixed point theorem.In Chapter6, we mention on impulsive fractional linear time-invariant (IF-LTI) system. Our main purpose is to obtain some necessary and sufficient conditions of controllability and observability for IF-LTI systems. As an ex-tension, the conclusions of controllability and observability for the continuous fractional LTI system are established.In Chapter7, the optimal control problem of a fractional dynamic system is investigated. A second order necessary optimality condition, which has not been discussed before, is given. An application is introduced to illustrate our main results.