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In this note our aim is to give a proof of the Pontryagin maximum principlefor a general optimal control problem with running state constraints and smoothdynamics. Our proof is based on the classical Ekeland variational principle.The main result (and its proof) of this note are not new and are alreadywell-known in the literature. The aim of the author is only to provide acomplete and detailed proof of this classical theorem in the case of smoothdynamics. If you have any remarks or questions, do not hesitate to contact theauthor at loic.bourdin@unilim.fr.

Note on Pontryagin maximum principle with running state constraints and smooth dynamics -- Proof based on the Ekeland variational principle