Trajectories Satisfying a State Constraint: Improved Estimates and New Non-Degeneracy Conditions

For a state-constrained control system described by a differential inclusion and a single functional inequality state constraint, it is known that, under an “inward pointing condition”, the W^1,1 distance of an arbitrary state trajectory to the set of state trajectories, which have the same left endpoint and which satisfy the state constraint, is linearly related to the state constraint violation. In this paper we show that, in situations where the state-constrained control system is described instead by a controlled differential equation, this estimate can be improved by replacing the W^1,1 distance on state trajectories by the Ekeland metric of the distance of the control functions. A counter-example reveals that a refinement of this nature is not in general valid for state constrained differential inclusions. Finally we show how the refined estimates may be used to establish new conditions for non-degeneracy of the state constrained Maximum Principle, in circumstances when the data depends discontinuously on the control variable.