VALUE FUNCTION AND OPTIMALITY CONDITIONS FOR SEMILINEAR CONTROL-PROBLEMS

This paper is concerned with optimal control problems of Mayer and Bolza type for systems governed by a semilinear state equation x'(t) = Ax(t) + f(t, x(t), u(t)), U(t) is-an-element-of U, where A is the infinitesimal generator of a strongly continuous semigroup in a Banach space X. We prove necessary and sufficient conditions for optimality and then use these conditions to investigate properties of the value function related to superdifferentials. Conversely, we use the value function to obtain criteria for optimality and feedback systems.