S. Shekhar, LOCAL CHARACTERIZATION OF INVARIANT-SETS OF AN AUTONOMOUS DIFFERENTIAL INCLUSION - BOUNDARY OF UNSTABLE MANIFOLDS, Journal of nonlinear science, 6(2), 1996, pp. 105-138
An application in robotics motivates us to characterize the evolution
of a subset in state space due to a compact neighborhood of an arbitra
ry dynamical system-an instance of a differential inclusion. Earlier r
esults of Blagodat-skikh and Filippov (1986) and Butkovskii (1982) cha
racterize the boundary of the attainable set and the forward projectio
n 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 different
ial inclusion contains an equilibrium point be called a singular invar
iant set. We show that the fields at the boundary of the forward proje
ction of a singular invariant set are degenerate under some regularity
assumptions when the state-wise boundary of the differential inclusio
n 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 for
ward projection of a singular invariant set. They characterize the bun
dle of unstable and stable manifolds of such a differential inclusion.