Optimal control and  approximate controllability for second-order integro-differential equations with state-dependent delay and non-instantaneous impulses

This article concerns the optimal control and the approximate controllability for second order integro-differential equation with state-dependent delay and non-instantaneous impulses. We first establish the existence of mild solution for the control system. Then, based on these results, we investigate the approximate controllability and show the existence of optimal controls for Bolza problems by using the resolvent family of linear operators, M"{o}nch's fixed point theorem, and the resolvent condition. Finally, we give an example to illustrate the effectiveness of the results. For more information see https://ejde.math.txstate.edu/Volumes/2025/54/abstr.html