Hemivariational inequalities and controllability results for second-order non-autonomous evolution system with impulsive effects

This research investigates the crucial concept of approximate controllability for a class of complex dynamical systems: second-order neutral non-autonomous evolution systems in Hilbert spaces. These systems, characterized by inherent memory effects (due to the neutral term) and non-smooth behavior (modeled by hemivariational inequalities), pose significant analytical challenges. We begin by rigorously establishing the existence of mild solutions for this intricate system. This crucial step relies on a powerful combination of mathematical tools, including cosine functions, a robust fixed-point technique, and the generalized Clarke’s subdifferential, which effectively handles the non-smoothness arising from the hemivariational inequalities. Building upon this foundation, we delve into the core objective: approximate controllability. This fundamental property explores the system’s ability to be arbitrarily close to any desired target state through judicious selection of control inputs. We derive sufficient conditions for approximate controllability, providing valuable insights into the system’s controllability characteristics. Finally, to underscore the practical significance of our theoretical findings, we present a concrete application demonstrating the developed theory’s efficacy in addressing real-world problems.