Sufficient conditions for optimal control problems with mixed constraints

In this paper we derive sufficient conditions for optimal control problems with mixed control and state constraints by applying a dual approach to the dynamic programming. These conditions guarantee that a relative minimum is achieved. We seek an optimal pair in the class of those admissible pairs for which graphs of trajectories belong to a set defined by a function satisfying a dual equation of dynamic programming. This set usually differs from the classical neighbourhood of an optimal trajectory. We prove a verification theorem and introduce an optimal dual feedback control.