We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the acceleration. We compute the relaxation of the problem which involves the so-called Fisher–Rao functional, a convex functional on the space of measures. This relaxation enables the derivation of several optimality conditions and, in particular, a sufficient condition which guarantees that a given path of the initial problem is also a minimizer of the relaxed one. Based on these sufficient conditions, the main result is that, when the value of the (minimized) functional is small enough, the minimizers are classical, that is the defect measure vanishes.
Accepté le :
DOI : 10.1051/cocv/2018068
Mots-clés : Riemannian cubics, splines, right-invariant metric on group of diffeomorphisms, relaxation
@article{COCV_2019__25__A70_0,
author = {Tahraoui, Rabah and Vialard, Fran\c{c}ois-Xavier},
title = {Minimizing acceleration on the group of diffeomorphisms and its relaxation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018068},
mrnumber = {4036657},
zbl = {1439.49024},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018068/}
}
TY - JOUR AU - Tahraoui, Rabah AU - Vialard, François-Xavier TI - Minimizing acceleration on the group of diffeomorphisms and its relaxation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018068/ DO - 10.1051/cocv/2018068 LA - en ID - COCV_2019__25__A70_0 ER -
%0 Journal Article %A Tahraoui, Rabah %A Vialard, François-Xavier %T Minimizing acceleration on the group of diffeomorphisms and its relaxation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018068/ %R 10.1051/cocv/2018068 %G en %F COCV_2019__25__A70_0
Tahraoui, Rabah; Vialard, François-Xavier. Minimizing acceleration on the group of diffeomorphisms and its relaxation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 70. doi : 10.1051/cocv/2018068. http://www.numdam.org/articles/10.1051/cocv/2018068/
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