Approximate controllability of second-order impulsive integro-differential systems involving state-dependent delay and damping effects via resolvent operator

Abstract

This paper investigates the controllability of a second-order impulsive damped integrodifferential nonautonomous system with state-dependent delay. The results are established using the properties of resolvent operators related to the second-order system behind them, and the analysis employs key mathematical tools such as Gronwall’s lemma and fixed-point theorems. Assuming the approximate controllability of the corresponding linearized system, we obtain a new set of sufficient conditions developed for the approximate controllability of the nonlinear second-order system. These findings contribute to our understanding of the approximate controllability in complex dynamical systems affected by both damping and state-dependent delay. Finally, we present an application that demonstrates and validates the theoretical findings of this study.