Geodesic
Optimal design of boundary observers for the wave equation
Pierre Jounieaux1 ; Yannick Privat2 ; Emmanuel Trélat3
1Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
2CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
3Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris, France
ESAIM. Proceedings, Tome 45 (2014), pp. 475-484
Cet article a éte moissonné depuis la source EDP Sciences

Voir la notice de l'article

In this article, we consider the wave equation on a domain of Rn with Lipschitz boundary. For every observable subset Γ of the boundary ∂Ω (endowed with the usual Hausdorff measure Hn − 1 on ∂Ω), the observability constant provides an account for the quality of the reconstruction in some inverse problem. Our objective is here to determine what is, in some appropriate sense, the best observation domain. After having defined a randomized observability constant, more relevant tan the usual one in applications, we determine the optimal value of this constant over all possible subsets Γ of prescribed area Hn − 1(Γ) = LHn − 1(∂Ω), with L ∈ (0,1), under appropriate spectral assumptions on Ω. We compute the maximizers of a relaxed version of the problem, and then study the existence of an optimal set of particular domains Ω. We then define and study an approximation of the problem with a finite number of modes, showing existence and uniqueness of an optimal set, and provide some numerical simulations.
DOI : 10.1051/proc/201445049

Pierre Jounieaux  1   ; Yannick Privat  2   ; Emmanuel Trélat  3

1 Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
2 CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
3 Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris, France 
@article{EP_2014_45_a49,
     author = {Pierre Jounieaux and Yannick Privat and Emmanuel Tr\'elat},
     title = {Optimal design of boundary observers for the wave equation},
     journal = {ESAIM. Proceedings},
     pages = {475--484},
     year = {2014},
     volume = {45},
     doi = {10.1051/proc/201445049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201445049/}
}
TY  - JOUR
AU  - Pierre Jounieaux
AU  - Yannick Privat
AU  - Emmanuel Trélat
TI  - Optimal design of boundary observers for the wave equation
JO  - ESAIM. Proceedings
PY  - 2014
SP  - 475
EP  - 484
VL  - 45
UR  - http://geodesic.mathdoc.fr/articles/10.1051/proc/201445049/
DO  - 10.1051/proc/201445049
LA  - en
ID  - EP_2014_45_a49
ER  - 
%0 Journal Article
%A Pierre Jounieaux
%A Yannick Privat
%A Emmanuel Trélat
%T Optimal design of boundary observers for the wave equation
%J ESAIM. Proceedings
%D 2014
%P 475-484
%V 45
%U http://geodesic.mathdoc.fr/articles/10.1051/proc/201445049/
%R 10.1051/proc/201445049
%G en
%F EP_2014_45_a49
Pierre Jounieaux; Yannick Privat; Emmanuel Trélat. Optimal design of boundary observers for the wave equation. ESAIM. Proceedings, Tome 45 (2014), pp. 475-484. doi: 10.1051/proc/201445049

Cité par Sources :