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Zbl 0951.49007
Clarke, F.H.; Ledyaev, Yu.S.; Radulescu, M.L.
Approximate invariance and differential inclusions in Hilbert spaces.
(English)
[J] J. Dyn. Control Syst. 3, No.4, 493-518 (1997). ISSN 1079-2724; ISSN 1573-8698/e

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 2000:
*49J24 Optimal control problems with differential inclusions (existence)
34A60 ODE with multivalued right-hand sides
34G20 Nonlinear ODE in abstract spaces
34H05 ODE in connection with control problems
49K24 Optimal control problems with differential inclusions (nec./ suff.)

Keywords: approximate weak and strong invariance; viability; lower and upper Hamiltonians; proximal normal cone; proximal aiming; $\varepsilon$-trajectory; differential inclusion

Cited in: Zbl pre06136289

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