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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.

MSC:

49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
34A60 Ordinary differential inclusions
34G20 Nonlinear differential equations in abstract spaces
34H05 Control problems involving ordinary differential equations
49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)
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