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

Published online by Cambridge University Press:  31 March 2010

Marcus Wagner
Marcus Wagner*
Affiliation:
University of Graz, Institute for Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria. www.thecitytocome.de; marcus.wagner@uni-graz.at

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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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.

Keywords

Quasiconvex functions with infinite valueslower semicontinuous quasiconvex envelopemultidimensional control problemrelaxationexistence of global minimizersimage registrationpolyconvex regularization

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011, pp. 190 - 221
Copyright
© EDP Sciences, SMAI, 2010

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