Trajectories of differential inclusions with state constraints

The paper deals with solutions of a differential inclusion constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure co F satisfies a strict inward pointing condition at every boundary point x∈∂Ω. Given a reference trajectory x⁎(⋅) taking values in an ε-neighborhood of Ω, we prove the existence of a second trajectory x:[0,T]↦Ω which satisfies . As shown by an earlier counterexample, this bound is sharp.