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Control problems for nonlocal set evolutions

Lorenz, Thomas

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Abstract

In this paper, we extend fundamental notions of control theory to evolving compact subsets of the Euclidean space. Dispensing with any restriction of regularity, shapes can be interpreted as nonempty compact subsets of the Euclidean space. Their family, however, does not have any obvious linear structure, but in combination with the popular Pompeiu-Hausdorff distance, it is a metric space. Here Aubin's framework of morphological equations is used for extending ordinary differential equations beyond vector spaces, namely to the metric space of nonempty compact subsets of the Euclidean space supplied with Pompeiu-Hausdorff distance. Now various control problems are formulated for compact sets depending on time: open-loop, relaxed and closed-loop control problems – each of them with state constraints. Using the close relation to morphological inclusions with state constraints, we specify sufficient conditions for the existence of compact-valued solutions.

Item Type: Preprint
Date Deposited: 22. Jan 2008 13:13
Date: 2008
Faculties / Institutes: Service facilities > Interdisciplinary Center for Scientific Computing
Subjects: 510 Mathematics
Controlled Keywords: Nichtlineare Kontrolltheorie, Verallgemeinerte Differentialgleichung, Kompakte Menge, Mengenwertige Abbildung, Differentialinklusion, Nebenbeding
Uncontrolled Keywords: Open-loop control , closed-loop control , relaxed control problem , morphological equation
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