Existence Results Of Several Classes Of Differential Inclusions And Their Applications

The dissertation deals with the existence results for several classes of evolution inclusions and their applications in the control theory. Specifically, we study the following problems:existence of anti-periodic (extremal) solutions for a class of first order evolution inclusions; feedback control for a class of evolution hemivariational inequalities of parabolic type; existence of mild solutions and "Bang-Bang" principle for fractional semilinear differential inclusions; existence of weak solutions for an abstract second order nonlinear evolution inclusion with its principal part having a small parameter ε and the asymptotic behavior of a sequence of its solutions when εâ†'0. 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.Chapter2concerns with some notations, definitions and preliminary facts needed in the following chapters, especially these from monotone operators theory, multivalued analysis, nonsmooth analysis and fractional calculus.In Chapter3, the existence of anti-periodic (extremal) solutions for a class of first order nonlinear evolution inclusions is considered. We study the problems under both convexity and nonconvexity conditions on the multivalued right-hand side. The main tools used here are the maximal monotone property of the derivative operator with anti-periodic conditions, the theory of L-pseudomonotone operators and some selection theorems from multivalued analysis. We remark that our approach is also applicable to the problem of anti-periodic solution for hemivariational inequalities of parabolic type.Chapter4is devoted to a control system governed by a class of evolution hemivariational inequalities. The constraint on the control is a multivalued map with nonconvex values which is lower semicontinuous with respect to the state variable. Meanwhile, we handle the same system in which the constraint on the control is the upper semicontinuous convex valued regularization of the original constraint. We obtain the existence results for the control systems and the relaxation property between the solution sets of these systems.In Chapter5, firstly, we consider the existence of mild solutions for fractional semilinear differential inclusions involving a nonconvex valued multifunction F(t,x(t)). Secondly, we study the relations between the solution sets of the original inclusion and the inclusions with the multivalued term taking the forms of co F(t,x(t)) and extco F(t,x(t)), respectively. Finally, a nonconvex optimal control problem described by fractional semilinear differential equations is considered.In Chapter6, we deal with an abstract second order nonlinear evolution inclusion with its principal part having a small parameter ε. We prove the existence of weak solution when the nonlinearity F being convex as well as nonconvex valued. Then we study the asymptotic behavior of a sequence of solutions{uε} when εâ†'0. We prove that there exists a limit function u and u is a solution of the corresponding first order evolution inclusion.