The primary objective of this manuscript is to develop a systematic framework for analysing feedback control systems governed by fractional differential equations involving the Caputo derivative of order \(1< \mu < 2\) in Banach spaces. Initially, the existence of mild solutions is established through fractional calculus, cosine operator families, and Schauder’s fixed-point theorem. Subsequently, by employing the Filippov theorem and the Cesari property, we derive a set of novel conditions that guarantee the existence of feasible control-state pairs for the system. To identify the optimal feedback control pairs, the main theoretical findings are systematically applied. Finally, a theoretical example is presented to illustrate the applicability and validate the proposed methodology and results.
