Relative Controllability of Nonlinear Implicit Fractional Integro-Differential Systems in Banach Spaces with Distributive Delays in the Control

In this work, Nonlinear Implicit Fractional Integrodifferential Systems in Banach Spaces with Distributed Delays in the Control were presented for Relative Controllability analysis. General argument was found which was used to establish the relationship between the relative controllability and the intersection of the two compact and convex set functions derived from the mild solution of the system. The establishment of the relationship gives impetus to the existence of  optimal control for the system as it becomes self- evidence that the intersection of the two compact and  convex set functions be non-void to establish relative controllability. Thus we have established relative controllability of our system. Uses were made of the notion of the measure of non-compactness of a set and Dabos' fixed point theorem, as well as the unsymmetric Fubinis' theorem to establish the mild solution of the system .Necessary and sufficient conditions for the existence of computable criterion for the relative controllability of our system were established. The establishment was built on the usage of definition of properness of the system and the effects of the existence of zero in the interior of a reachable set of any dynamical control system.

Keywords:- Maximization, Dabos' Fixed Point Theorem, Mild Solution, Optimal Control Relative Controllability, Set Function, Measure of Non-Compactness, Calculus of Variation.