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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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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 
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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     title = {Quasiconvex relaxation of multidimensional control problems with integrands $f(t,\xi ,v)$},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {190--221},
     year = {2011},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     doi = {10.1051/cocv/2010008},
     zbl = {1217.49007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010008/}
}
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