Abstract
Consider a mappingF from a Hilbert spaceH to the subsets ofH, 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 subsetS ofH 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 toF 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.
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Dedicated to Professor R. V. Gamkrelidze
The first and the third authors are supported in part by the Natural Sciences and Engineering Research Council of Canada and Le Fonds FCAR du Québec.
The second author is visiting Rutgers University. Supported in part by the Russian Foundation for Fundamental Research, Grant 96-01-00219, by the Natural Sciences and Engineering Research Council of Canada, Le Fonds FCAR du Québec, and by the Rutgers Center for Systems and Control (SYCON).
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Clarke, F.H., Ledyaev, Y.S. & Radulescu, M.L. Approximate invariance and differential inclusions in Hilbert spaces. Journal of Dynamical and Control Systems 3, 493–518 (1997). https://doi.org/10.1007/BF02463280
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DOI: https://doi.org/10.1007/BF02463280