Maximum principle for a control problem governed by an evolution equation

We prove the maximum principle for an optimal control problem governed by t he system y'(t) + A(t)y(t) = f (t, y(t), u(t)), u(t) epsilon U(t), with state constraint (y(0),y(T)) epsilon C subset of H x H, under three di fferent hypotheses: (H1) C is a convex set with nonempty interior; (H2) C = {y(0)} x C-0, with C-0 a convex set with nonempty interior in H and the ev olution system satisfying compactness hypotheses; (H3) the periodic case y( 0) = y(T), with the evolution system satisfying compactness hypotheses. We do not assume the controls to be bounded. We give some examples for distrib uted control problems.