Controllability And Applications Of Some Fractional Differential Systems

The dissertation deals with the controls and applications for some fractional equations evolution. We study the following problems:the Solvability and optimal controls for some fractional impulsive equa-tion; the solvability and optimal controls for some fractional impulsive equations of order1<α<2; approximate controllability of impulsive fractional neutral evolution equations with Riemann-Liouville frac-tional derivatives; on the approximate controllability for some impul-sive fractional evolution hemivariational inequalities, at last,we study dynamic analysis of some impulsive fractional-order neural network with mixed delay. This dissertation consists of6chapters. They are organized as follows:In Chapter1, we introduce the background of our research, the context of this dissertation and the main results obtained.In Chapter2, we show some preliminaries such as definitions needed in the following chapters, especially these come from functional analysis, multivalued analysis, semigroup theory, generalized gradient and fractional calculus.In Chapter3, by the Gronwall inequality and Leray-Schauder fixed point theorem, we proved the existence and uniqueness of the mild solution of some fractional impulsive equation, we also use anal-ysising Lagrange function to verify optimal controls. We also concern about some fractional impulsive equations of order1<α<2. With a view to the impulsive item, we deduce the integral formula. Follow the given condition, by the semigroup theory and fixed point theo-rem, we obtain the solvability results, at last we also verify optimal controls. In Chapter4, we consider impulsive fractional neutral evolution equations with Riemann-Liouville fractional derivatives. Firstly, we define the corresponding space and deduce the integral formula. Under the given condition, based on semigroup theory, we talk about the existence and uniqueness of mild solution of the equation. At last, using the Nemytskii operator, the approximate controllability is also verified.In Chapter5, we deal with some impulsive fractional evolution hemivariational inequalities. Under the condition that the F is convex and upper semicontinuous, we prove the existence of the mild solution. If the multifunction F is measurable and weak converge, we also prove that the system is approximate controllable.In Chapter6, we analysis some impulsive fractional-order neural network with mixed delay. We deduce the integral formula and show the stability of the system. At last, some examples is given to illustrate the result.