  We provide new infinitesimal characterizations for strong invariance of
multifunctions in terms of Hamiltonian inequalities and tangent cones. In lieu
of the standard local Lipschitzness assumption on the multifunction, we assume
a new feedback realizability condition that can in particular be satisfied by
control systems that are discontinuous in the state variable. Our realization
condition is based on H. Sussmann's unique limiting property, and allows a more
general class of feedback realizations than is allowed by the recent strong
invariance characterizations of Krastanov, Malisoff, and Wolenski. We also give
new nonsmooth monotonicity characterizations for control systems that may be
discontinuous in the state.
