Stratified necessary conditions for unbounded differential inclusions with state constraints

The concept of stratified necessary conditions for an optimal control problem, whose dynamic constraint is formulated as a differential inclusion, was recently introduced by F. H. Clarke. These are conditions satisfied by a (feasible) state trajectory that achieves the minimum value of a cost function, not over all state trajectories, but just ones whose velocities lie in a time-varying open ball of specified radius about the velocity of the state trajectory of interest. Considering different radius functions stratifies the interpretation of `minimizer´. In this paper we announce extensions of currently available stratified necessary conditions, to allow for the presence of a unilateral state constraint. As was shown by Clarke in the state constraint-free case, we find that, also in our more general setting, the stratified necessary conditions yield generalizations of earlier necessary conditions for unbounded differential inclusions as simple corollaries.