eprintid: 8022
rev_number: 13
eprint_status: archive
userid: 1
dir: disk0/00/00/80/22
datestamp: 2008-01-22 13:13:26
lastmod: 2015-04-21 13:39:33
status_changed: 2012-08-14 15:24:01
type: preprint
metadata_visibility: show
creators_name: Lorenz, Thomas
title: Control problems for nonlocal set evolutions
ispublished: pub
subjects: ddc-510
divisions: i-708000
keywords: Open-loop control , closed-loop control , relaxed control problem , morphological equation
cterms_swd: Nichtlineare Kontrolltheorie
cterms_swd: Verallgemeinerte Differentialgleichung
cterms_swd: Kompakte Menge
cterms_swd: Mengenwertige Abbildung
cterms_swd: Differentialinklusion
cterms_swd: Nebenbeding
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. 
abstract_translated_lang: eng
class_scheme: msc
class_labels: 37N35, 34A60, 49J24, 93B03, 93B52
date: 2008
date_type: published
id_scheme: DOI
id_number: 10.11588/heidok.00008022
ppn_swb: 1646175964
own_urn: urn:nbn:de:bsz:16-opus-80224
language: eng
bibsort: LORENZTHOMCONTROLPRO2008
full_text_status: public
citation:   Lorenz, Thomas  (2008) Control problems for nonlocal set evolutions.  [Preprint]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/8022/1/Lorenz_morphological_control_problems.pdf.old
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/8022/2/Lorenz_morphological_control_problems_revised.pdf