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Nonsmooth shape evolutions under state constraints: A viability theorem

Lorenz, Thomas

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Abstract

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space). In this morphological framework, the evolution of compact subsets of the Euclidean space is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis. Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order) - correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space. We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in the Euclidean space, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set M.

Item Type: Preprint
Series Name: IWR-Preprints
Date Deposited: 22. Nov 2006 13:06
Date: 2006
Faculties / Institutes: Service facilities > Interdisciplinary Center for Scientific Computing
Subjects: 510 Mathematics
Controlled Keywords: Verallgemeinerte Differentialgleichung, Kompakte Menge, Mengenwertige Abbildung, Nebenbedingung
Uncontrolled Keywords: Shape evolutions with constraints , velocity method (speed method) , morphological equations , Nagumo's theorem , viability condition
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