Existence of Optimal Feedback Control via Hilfer Derivative for Semilinear Fractional Evolution Equations: Theoretical Insights and Applications

In this research, we delve into the optimal feedback controls (OFCs) of a system employing a compact semigroup in Banach spaces. The system is governed by semilinear fractional evolution equations (FEEs) of order [Formula: see text]. We demonstrate the existence of feasible pairs by utilizing the Cesari property, Arzela Ascoli, the Fillippove theorem, and an extension of previous work on FEEs. We provide a finding about the presence of ideal control pairs in relation to the Lagrange issue. Finally, we provide real-world applications and an example.