Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 54, pp. 1-18.

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

Authors: Abdelhamid Bensalem (Djillali Liabes Univ. of Sidi Bel-Abbes, Algeria)
         Abdelkrim Salim  (Djillali Liabes Univ. of Sidi Bel-Abbes, Algeria)
         Mouffak Benchohra (Djillali Liabes Univ. of Sidi Bel-Abbes, Algeria)
         Gaston M. N'Guerekata (Morgan State Univ., Baltimore, MD, USA)

Abstract:
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.


Submitted March 3, 2025. Published May 26, 2025.
Math Subject Classifications: 93B05, 49N25, 45J05, 47H08, 34K45.
Key Words: Approximate controllability; optimal control; fixed point theorem;
infinite delay; second-order integrodifferential equation;
impulsive; measures of noncompactness; mild solution; resolvent operator.