This paper investigates a class of fractional semilinear integro-differential control systems of order \(\alpha \in (1,2)\) in Hilbert spaces, with dynamics governed by the Caputo fractional derivative. Utilizing resolvent operator theory, we establish sufficient conditions for the existence and uniqueness of mild solutions through the Banach fixed point theorem. Under appropriate assumptions, the existence of at least one optimal control pair for the associated Lagrange problem is demonstrated. To illustrate the applicability and effectiveness of the theoretical results, several examples are presented.
