Optimal Control Results for Fractional Nonlocal Integro-Differential Systems of Order (1,2) via Resolvent Operators

This work studies a class of fractional nonlocal semilinear integro-differential control systems of order α ∈ (1,2) in Hilbert spaces. The dynamics are described by Caputo derivatives with nonlocal initial conditions. Using resolvent operators for fractional evolution equations, we provide sufficient conditions for the existence and uniqueness of mild solutions through the Banach fixed point theorem. The analysis assumes Lipschitz continuity and linear growth of the nonlinear term, boundedness of the associated operators, and admissibility of the control operator. An optimal control problem with a quadratic cost functional on a convex admissible control set is then considered. By applying minimizing sequence techniques, reflexivity of the control space, and weak lower semicontinuity arguments, the existence of an optimal control is established. An example for validation is included in the paper to further support the theoretical findings.