OPTIMALITY CONDITIONS FOR THE RELATIVE CONTROLLABILITY OF NEUTRAL FUNCTIONAL INTEGRODIFFERENTIAL SYSTEMS IN BANACH SPACES WITH DISTRIBUTED DELAYS IN THE CONTROL

In this paper, Neutral Functional Integrodifferential Systems in Banach Spaces with Distributed Delays in the Control was presented with the main objective of investigating the existence of an optimal control for the system . We used variation of parameters to obtain the solution of our system of interest as an integral equation. The integral equation contains the values of the control  .The values of the control u(t) for  enter into the definition of the initial complete state. To separate them we applied the Unsymmetric Fubini Theorem and the integration is in the Lebesque-Stielties sense. Use is also made of the Game of Pursuit to establish the existence of an optimal control of the system. The set functions (reachable set, attainable set, target set and controllability grammian) upon which our work hinges were extracted from the state. Necessary and sufficient conditions for the existence and form of optimal control for the Neutral Functional Integrodifferential Systems in Abstract Space (Banach Space) with Distributed Delays in the control were established.&nbsp