Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms

Thomas Lorenz

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2008)

  • Volume: 28, Issue: 1, page 15-73
  • ISSN: 1509-9407

Abstract

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Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space whose evolution is determined by the nonlocal properties of both the current set and the normal cones at its topological boundary. Indeed, simple deformations such as isotropic expansions exemplify that topological boundaries do not have to evolve continuously in time and thus Aubin's original concept cannot be applied directly. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. The regularity of compact reachable sets of differential inclusions is studied extensively instead. This example of nonlocal set evolutions in the Euclidean space serves as an introductory motivation for extending ordinary differential equations (and evolution equations) beyond the traditional border of vector spaces - and for combining it with other examples in systems.

How to cite

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Thomas Lorenz. "Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 28.1 (2008): 15-73. <http://eudml.org/doc/271152>.

@article{ThomasLorenz2008,
abstract = { Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space whose evolution is determined by the nonlocal properties of both the current set and the normal cones at its topological boundary. Indeed, simple deformations such as isotropic expansions exemplify that topological boundaries do not have to evolve continuously in time and thus Aubin's original concept cannot be applied directly. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. The regularity of compact reachable sets of differential inclusions is studied extensively instead. This example of nonlocal set evolutions in the Euclidean space serves as an introductory motivation for extending ordinary differential equations (and evolution equations) beyond the traditional border of vector spaces - and for combining it with other examples in systems. },
author = {Thomas Lorenz},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {mutational equations; quasidifferential equations; funnel equations; nonlocal geometric evolutions; reachable sets of differential inclusions; sets of positive erosion; sets of positive reach; Mutational equations; sets of positive erosion, sets of positive reach.},
language = {eng},
number = {1},
pages = {15-73},
title = {Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms},
url = {http://eudml.org/doc/271152},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Thomas Lorenz
TI - Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2008
VL - 28
IS - 1
SP - 15
EP - 73
AB - Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space whose evolution is determined by the nonlocal properties of both the current set and the normal cones at its topological boundary. Indeed, simple deformations such as isotropic expansions exemplify that topological boundaries do not have to evolve continuously in time and thus Aubin's original concept cannot be applied directly. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. The regularity of compact reachable sets of differential inclusions is studied extensively instead. This example of nonlocal set evolutions in the Euclidean space serves as an introductory motivation for extending ordinary differential equations (and evolution equations) beyond the traditional border of vector spaces - and for combining it with other examples in systems.
LA - eng
KW - mutational equations; quasidifferential equations; funnel equations; nonlocal geometric evolutions; reachable sets of differential inclusions; sets of positive erosion; sets of positive reach; Mutational equations; sets of positive erosion, sets of positive reach.
UR - http://eudml.org/doc/271152
ER -

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