Quasiconvex relaxation of multidimensional control problems with integrands $f(t,\xi ,v)$

Wagner, Marcus

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 190-221.

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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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

Classification : 26B05, 26B25, 49J20, 49J45, 68U10
Mots clés : 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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