Existence and Uniqueness of Optimal Cotrol of Impulsive Quasi-Linear Fractional Mixed Volterra–Fredholm- Type Integro - Differential Euations in Banach Spaces with Multiple Delays in the Control

Paul Anaetodike Oraekie

The Impulsive Quasi - Linear Fractional Mixed Volterra - Fredholm – Type Integro – Differential Equations in Banach Spaces with Multiple Delays in the Control is presented for controllability analysis. Necessary and Sufficient Conditions for the Existence of an Optimal Control for the System are established. The mild solution of the system is established using the variation of constant formula. The set functions upon which our study hinged were extracted. Necessary and sufficient conditions for the establishment of the uniqueness of the system were derived. Use was made of some controllability standards to establish results. The establishment of the uniqueness of the optimal control provided a new approach for the proof of the existence of an optimal control of any dynamical control system. The main result is built on the maximization of a set function; a technique drawn from the calculus of variation.