| Issue |
ESAIM: COCV
Volume 25, 2019
|
|
|---|---|---|
| Article Number | 70 | |
| Number of page(s) | 45 | |
| DOI | https://doi.org/10.1051/cocv/2018068 | |
| Published online | 27 November 2019 | |
Minimizing acceleration on the group of diffeomorphisms and its relaxation
1
UMR 6085 CNRS-Université de Rouen,
76801
Saint Etienne du Rouvray, France.
2
Université Paris-Est, LIGM (UMR 8049), UPEM,
F77454
Marne-la-Vallée, France.
* Corresponding author: francois-xavier.vialard@u-pem.fr
Received:
31
August
2017
Accepted:
3
December
2018
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.
Mathematics Subject Classification: 53-XX / 49-XX
Key words: Riemannian cubics / splines / right-invariant metric on group of diffeomorphisms / relaxation
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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