Optimal solutions to differential inclusions in presence of state constraints

Necessary conditions in terms of the Hamiltonian are given for optimal solutions to the differential inclusion problem when state constraints are present. This result extends a result of Clarke for the unconstrained problem. The data are nonsmooth, nonlinear, nonconvex. The method incorporates the state constraint in the cost functional as a penalty term for a sequence of unconstrained problems that approximate our problem. An application of Ekeland's variational principle, the known necessary conditions for the auxiliary problems, and a limiting process provide the necessary conditions.