Investigation on the approximate controllability of Sobolev-type second-order damped stochastic non-autonomous differential inclusions

This work investigates the approximate controllability of a class of damped second-order stochastic nonlinear differential inclusions of Sobolev-type governed by hemivariational inequalities in Hilbert spaces. To begin with, the existence of mild solutions for the considered second-order differential system is established by employing tools from non-smooth analysis, particularly the framework of the generalized Clarke's subdifferential. Next, sufficient conditions ensuring the existence of mild solutions are derived through a fixed point methodology for condensing multi-valued operators, in combination with properties of the generalized Clarke's subdifferential and strongly continuous cosine families. Building upon these analytical results, the approximate controllability of the system is subsequently verified. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the theoretical results obtained in this study.