Results on the optimal feedback control for second-order non-autonomous evolution equations with damping

In this paper, we investigate a class of optimal feedback control concerning second-order non-autonomous evolution equations with damping in Banach spaces. Initially, we determine the existence results of mild solution for these equations by utilising the Leray-Schauder's alternative fixed-point theorem and the Banach's fixed-point theorem, incorporating Lipschitz conditions and various forms of boundedness criteria. Subsequently, by utilising the Filippove theorem and the Cesari property, we introduce a novel set of sufficient assumptions are formulated to ensure the existence results of feasible pairs for the feedback control systems. Finally, an application is presented to highlight our primary findings.