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Approximate invariance and differential inclusions in Hilbert spaces. (English) Zbl 0951.49007
Summary: 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 $\varepsilon$-trajectory corresponding to a differential inclusion.

49J24Optimal control problems with differential inclusions (existence) (MSC2000)
34A60Differential inclusions
34G20Nonlinear ODE in abstract spaces
34H05ODE in connection with control problems
49K24Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)