Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r∈(1,2)

This article investigates the issue of optimal control and approximate controllability results for fractional integrodifferential evolution equations with infinite delay of (Formula presented.) in Banach space. In the beginning, we analyze approximate controllability results for fractional integrodifferential evolution equations using the fractional calculations, cosine families, and Banach fixed point theorem. After, we developed the continuous dependence of the fractional integrodifferential evolution equations by using the Henry–Gronwall inequality. Furthermore, we tested the existence of optimal controls for the Lagrange problem. Lastly, an application is presented to illustrate the theory of the main results.