Trajectories of Differential Inclusions with State Constraints

The paper deals with solutions of a differential inclusion ẋ ∈ F (x) constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure coF 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 ] 7→ Ω which satisfies ‖x−x‖W 1,1 ≤ Cε(1+ | ln ε|). As shown by an earlier counterexample, this bound is sharp.