Approximate Invariance and Differential Inclusions in Hilbert Spaces

Abstract

Consider a mapping F from a Hilbert space H to the subsets of H, which is upper semicontinuous/Lipschitz, has nonconvex, noncompact values, and satisfies a linear growth condition. We give the first necessary and sufficient conditions in this general setting for a subset S of H to be approximately weakly/strongly invariant with respect to approximate solutions of the differential inclusion \dot{x}(t) \in F(x). The conditions are given in terms of the lower/upper Hamiltonians corresponding to F and involve nonsmooth analysis elements and techniques. The concept of approximate invariance generalizes the well-known concept of invariance and in turn relies on the notion of an e-trajectory corresponding to a differential inclusion.