Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)

Marcus Wagner

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 190-221
  • ISSN: 1292-8119

Abstract

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We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.

How to cite

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Wagner, Marcus. "Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 190-221. <http://eudml.org/doc/272959>.

@article{Wagner2011,
abstract = {We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.},
author = {Wagner, Marcus},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {quasiconvex functions with infinite values; lower semicontinuous quasiconvex envelope; multidimensional control problem; relaxation; existence of global minimizers; image registration; polyconvex regularization},
language = {eng},
number = {1},
pages = {190-221},
publisher = {EDP-Sciences},
title = {Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)},
url = {http://eudml.org/doc/272959},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Wagner, Marcus
TI - Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 1
SP - 190
EP - 221
AB - We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.
LA - eng
KW - quasiconvex functions with infinite values; lower semicontinuous quasiconvex envelope; multidimensional control problem; relaxation; existence of global minimizers; image registration; polyconvex regularization
UR - http://eudml.org/doc/272959
ER -

References

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