Geodesic
Quasiconvex relaxation of multidimensional control problems with integrands f(t,ξ,v)
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 190-221

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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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

DOI : 10.1051/cocv/2010008
Classification : 26B05, 26B25, 49J20, 49J45, 68U10
Keywords: quasiconvex functions with infinite values, lower semicontinuous quasiconvex envelope, multidimensional control problem, relaxation, existence of global minimizers, image registration, polyconvex regularization 
@article{COCV_2011__17_1_190_0,
     author = {Wagner, Marcus},
     title = {Quasiconvex relaxation of multidimensional control problems with integrands $f(t,\xi ,v)$},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {190--221},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     doi = {10.1051/cocv/2010008},
     zbl = {1217.49007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010008/}
}
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Wagner, Marcus. Quasiconvex relaxation of multidimensional control problems with integrands $f(t,\xi ,v)$. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 190-221. doi: 10.1051/cocv/2010008

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