Résumé en anglais

We derive Pontryagin’s maximum principle for a general optimal control problem using the set-valued version of variational equation. We achieve this aim by exploiting an adequate differential calculus of set-valued maps. Furthermore, the calmness condition is replaced by a surjectivity condition involving reachable sets of the “set-valued linearization” of the initial control problem. Duality then provides both the “adjoint differential inclusion” and the maximum principle.