eprintid: 6972
rev_number: 13
eprint_status: archive
userid: 1
dir: disk0/00/00/69/72
datestamp: 2006-11-22 13:06:04
lastmod: 2015-04-27 04:36:22
status_changed: 2012-08-14 15:20:06
type: preprint
metadata_visibility: show
creators_name: Lorenz, Thomas
title: Nonsmooth shape evolutions under state constraints: A viability theorem
ispublished: pub
subjects: ddc-510
divisions: i-708000
keywords: Shape evolutions with constraints , velocity method (speed method) , morphological equations , Nagumo's theorem , viability condition
cterms_swd: Verallgemeinerte Differentialgleichung
cterms_swd: Kompakte Menge
cterms_swd: Mengenwertige Abbildung
cterms_swd: Nebenbedingung
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. 
abstract_translated_lang: eng
class_scheme: msc
class_labels: 34A60, 49J53, 49J24, 93C15, 49Q10
date: 2006
date_type: published
id_scheme: DOI
id_number: 10.11588/heidok.00006972
portal_cluster_id: p-iwrpp
portal_order: 06972
ppn_swb: 1646175476
own_urn: urn:nbn:de:bsz:16-opus-69728
language: eng
bibsort: LORENZTHOMNONSMOOTHS2006
full_text_status: public
series: IWR-Preprints
citation:   Lorenz, Thomas  (2006) Nonsmooth shape evolutions under state constraints: A viability theorem.  [Preprint]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/6972/1/Lorenz_shape_evolutions_under_constraints_JMAA.pdf
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/6972/2/Lorenz_viability_using_transitions_in_LIP.pdf.old