Optimal design of boundary observers for the wave equation
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 qua...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
EDP Sciences
2014-09-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | http://dx.doi.org/10.1051/proc/201445049 |
| _version_ | 1797971375040757760 |
|---|---|
| author | Jounieaux Pierre Privat Yannick Trélat Emmanuel |
| author_facet | Jounieaux Pierre Privat Yannick Trélat Emmanuel |
| author_sort | Jounieaux Pierre |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T03:31:37Z |
| format | Article |
| id | doaj.art-ccc33819732d4e45ba8d36829c479d97 |
| institution | Directory Open Access Journal |
| issn | 2267-3059 |
| language | English |
| last_indexed | 2024-04-11T03:31:37Z |
| publishDate | 2014-09-01 |
| publisher | EDP Sciences |
| record_format | Article |
| series | ESAIM: Proceedings and Surveys |
| spelling | doaj.art-ccc33819732d4e45ba8d36829c479d972023-01-02T06:33:47ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592014-09-014547548410.1051/proc/201445049proc144549Optimal design of boundary observers for the wave equationJounieaux Pierre0Privat Yannick1Trélat Emmanuel2Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis LionsCNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis LionsSorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de FranceIn 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.http://dx.doi.org/10.1051/proc/201445049 |
| spellingShingle | Jounieaux Pierre Privat Yannick Trélat Emmanuel Optimal design of boundary observers for the wave equation ESAIM: Proceedings and Surveys |
| title | Optimal design of boundary observers for the wave equation |
| title_full | Optimal design of boundary observers for the wave equation |
| title_fullStr | Optimal design of boundary observers for the wave equation |
| title_full_unstemmed | Optimal design of boundary observers for the wave equation |
| title_short | Optimal design of boundary observers for the wave equation |
| title_sort | optimal design of boundary observers for the wave equation |
| url | http://dx.doi.org/10.1051/proc/201445049 |
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