Relaxed Trajectories of Integrodifferential Equations and Optimal Control on Banach Space

Relaxation optimal control for a class of integrodifferential equation associated with fractional power operator is investigated. By Gronwall lemma with singularity and delay the existence of mild solutions of integrodifferential equations is proved. Introducing finitely additive measure control, we convexificate original control systems and obtain corresponding relaxed control systems. Approximation results and equivalent relationship between control differential equations and integrodifferential inclusion are given. Existence of relaxed optimal control is verified and one relaxed optimal control is just an original one under generalized Cesari conditions.