Abstract
In this paper we examine optimal control problems governed by nonlinear evolution equations. First we establish the existence of optimal controls under a convexity hypothesis on an appropriate orientor field. Then we drop the convexity hypothesis. Now in order to guarantee the existence of optimal solutions we need to pass to a larger system known as the "relaxed system". We introduce that system, show that it has optimal solutions and compare it with original one. Finally using the Dubovitsky-Milyutin formalism we obtain necessary optimality conditions (maximum principle) and we prove a new "bang-bang" theorem.
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