A strong version of the maximum principle under weak hypotheses

We present a version of the finite-dimensional maximum principle that incorporates high-order point variations and is valid under minimal technical assumptions, weaker than those of the usual classical and nonsmooth versions. The proof follows the classical idea of using needle variations and applying an appropriate open mapping theorem to a multiparameter variation, whose effect is computed in terms of those of the needle variations by means of the chain rule. However, this has to be carried out in a new setting, namely, the class of “semidifferentiable maps”, that contains all maps arising in the optimal control problem and has a concept of generalized differential with all the right properties