A weak maximum principle for control problems with state constraints

New first-order necessary conditions of optimality for control problems with state constraints are provided. These conditions are a variant of the nonsmooth maximum principle in which an Euler-Lagrange inclusion is involved. The main novelty of the result is precisely the ability to address state constraints, generalizing a known Euler-Lagrange inclusion for optimal control problems. The conditions developed are, in some cases, stronger than the standard maximum principle, since they can reduce the set of candidates to minimizers.