NECESSARY CONDITIONS FOR FUNCTIONAL-DIFFERENTIAL INCLUSIONS

We consider an optimal control problem in which the dynamic equation a nd cost function depend on the recent past of the trajectory. The regu larity assumed in the basic data is Lipschitz continuity with respect to the sup norm. It is shown that, for a given optimal solution, an ad joint are of bounded variation exists that satisfies an associated Ham iltonian inclusion. From this result, known smooth versions of the Pon tryagin maximum principle for hereditary problems can be easily derive d. Problems with Euclidean endpoint constraints are also considered.