Approximate Invariance and Differential Inclusions in Hilbert Spaces

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\dot x(t) \in F(x)$$ \end{document} 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 ∈-trajectory corresponding to a differential inclusion.