Quasiconvex relaxation of multidimensional control problems with integrands f(t,ξ,v)
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Submission date: 17. Oct. 2008
published in: Control, optimisation and calculus of variations (ESAIM-COCV), 17 (2011) 1, p. 190-221
DOI number (of the published article): 10.1051/cocv/2010008
MSC-Numbers: 26B05, 26B25, 49J20, 49J45, 68U10
Keywords and phrases: Quasiconvex functions with infinite values, lower semicontinuous quasiconvex envelope, multidimensional control problem, relaxation, existence of global minimizers, image registration, polyconvex regularization
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We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands 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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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.