Local Characterization of Invariant Sets of an Autonomous Differential Inclusion: Boundary of Unstable Manifolds

An application in robotics motivates us to characterize the evolution of a subset in state space due to a compact neighborhood of an arbitrary dynamical system - an instance of a differential inclusion. Earlier results of Blagodat-skikh and Filippov (1986) and Butkovskii (1982) characterize the boundary of the attainable set and the forward projection operator of a state. Our first result is a local characterization of the boundary of the forward projection of a compact regular subset of the state space. Let the collection of states such that the differential inclusion contains an equilibrium point be called a singular invariant set. We show that the fields at the boundary of the forward projection of a singular invariant set are degenerate under some regularity assumptions when the state-wise boundary of the differential inclusion is smooth. Consider instead those differential inclusions such that the state-wise boundary of the problem is a regular convex polytope - a piecewise smooth boundary rather than smooth. Our second result gives conditions for the uniqueness and existence of the boundary of the forward projection of a singular invariant set. They characterize the bundle of unstable and stable manifolds of such a differential inclusion.