On Some Boundary Value Problems for Fractional Feedback Control Systems

Abstract

We consider a nonlocal boundary value problem for a feedback control system governed by a fractional degenerate (Sobolev type) semilinear differential inclusion in a Banach space. To solve this problem, we introduce a multivalued integral operator whose fixed points determine its solutions and study the properties of this operator. It is demonstrated, in particular, that the operator is condensing with respect to an appropriate measure of noncompactness in a functional space. This makes it possible to formulate a general existence principle (Theorem 33) in terms of the topological degree theory. Theorem 34 gives an example of a concrete realization of this principle. As its corollary we get the existence of an optimal solution to our problem (Theorem 35). Some important particular cases including a nonlocal Cauchy problem, periodic and anti-periodic boundary value problems are presented. As example, we consider the existence of an optimal periodic solution for a fractional diffusion type degenerate control system.