An analysis on the optimal control and approximate controllability for Hilfer fractional neutral integro-differential systems with finite delay

The existence and uniqueness of solutions to Hilfer fractional neutral delay integro-differential equations subject to nonlocal conditions are discussed in this study. The main results of this study are based in part on the fixed point techniques of Banach contraction and Krasnoselskii's fixed point theorem from calculus theory. First, we determine whether or not the fractional system has a mild solution. The uniqueness of the mild solution is further illustrated by expanding our results. The optimal control problems are governed by a new class of neutral delay integro-differential equations in Banach spaces, and we also develop sufficient conditions for the approximate controllability of a nonlinear fractional system. An example is given at the end to strengthen the compatibility of the results.