Properties of the Set of Admissible “State Control” Pair for a Class of Fractional Semilinear Evolution Control Systems

In this paper, we discuss a class of Caputo fractional evolution equations on Banach space with feedback control constraint whose value is non-convex closed in the control space. First, we prove the existence of solutions for the system with feedback control whose values are the extreme points of the convexified constraint that belongs to the original one. Secondly, we study the topological properties of the sets of admissible “state-control” pair for the original system with various feedback control constraints and the relations between them. Moreover, we obtain necessary and sufficient conditions for the solution set of original systems to be closed. In the end, an example is given to illustrate the applications of our main results.