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]→Ω which satisfies ||x-x*||W^1,1{less-than above slanted equal above greater-than above slanted equal}Cε(1+|lnε|). As shown by an earlier counterexample, this bound is sharp.