An Euler-Lagrange inclusion for optimal control problems with state constraints

New first-order necessary conditions for optimality for control problems with pathwise state constraints are given. These conditions are a variant of a nonsmooth maximum principle which includes a joint subdifferential of the Hamiltonian - a condition called Euler-Lagrange inclusion (ELI). The main novelty of the result provided here is the ability to address state constraints while using an ELI. The ELI conditions have a number of desirable properties. Namely, they are, in some cases, able to convey more information about minimizers, and for the normal convex problems they are sufficient conditions of optimality. It is shown that these strengths are retained in the presence of state constraints.