Existence theorems are considered for relaxed optimal control problems described by semilinear systems in Banach spaces. Relaxed controls are used whose values are finitely additive probability measures; this class of relaxed controls does not require special assumptions (such as compactness) on the control set. Under suitable conditions, relaxed trajectories coincide with those obtained from differential inclusions. Existence theorems for relaxed controls are obtained that apply to distributed parameter systems described by semilinear parabolic and wave equations, as well as a version of Pontryagin’s maximum principle for relaxed optimal control problems.