Local characterization of invariant sets of an autonomous differential inclusion

Summary 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 theattainable set and theforward projection operator of a state. Our first result is a local characterization of the boundary of the forward projection ofa compact regular subset of the state space. Let the collection of states such that the differential inclusion contains an equilibrium point be called asingular 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 theuniqueness andexistence 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.