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A strong maximum principle for systems of differential inclusions

We present a version of the finite-dimensional maximum principle for systems of differential inclusions, that contains as particular cases of previous versions for control systems as well as for a single differential inclusion. The result incorporates high-order point variations and is valid under minimal technical assumptions, weaker than those of most 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. We reduce the differential inclusions case to the vector field system case. The key technical tool enabling us to carry out the reduction which is a generalization of a selection theorem due to Bressan (1988).