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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.18976 |
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| _version_ | 1866915559026720768 |
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| author | Dehman, Belhassen Zuazua, Sylvain Ervedoza an Enrique |
| author_facet | Dehman, Belhassen Zuazua, Sylvain Ervedoza an Enrique |
| contents | We establish sharp regional observability results for solutions of the wave equation in a bounded domain of $Ω\subset \mathbb{R}^n$, in case where the geometric control condition is not satisfied. Assuming that the waves are observed on a non-empty open subset $ω\subset Ω$ and that the initial data are supported in another open subset $\mathscr{O} \subsetΩ$, we derive estimates for the energy of initial data localized in $\mathscr{O}$, in terms of the energy measured on the observation set $(0,T) \times ω$. This holds under a suitable geometric condition relating the time horizon T and the subdomains $ω$ and $\mathscr{O}$. By duality, we obtain new controllability results for the wave equation, ensuring that the projection of the solution onto $\mathscr{O}$ can be controlled by means of controls supported in $ω$, with optimal spatial support. We also present several extensions of the main result, including the case of boundary observations, as well as a characterization of the observable fraction of the energy of the initial data from partial measurements on $(0,T) \times ω$. Applications to wave control are discussed accordingly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regional and partial observability and control of waves Dehman, Belhassen Zuazua, Sylvain Ervedoza an Enrique Analysis of PDEs 35A18, 35Lxx, 35Q93, 93Dxx We establish sharp regional observability results for solutions of the wave equation in a bounded domain of $Ω\subset \mathbb{R}^n$, in case where the geometric control condition is not satisfied. Assuming that the waves are observed on a non-empty open subset $ω\subset Ω$ and that the initial data are supported in another open subset $\mathscr{O} \subsetΩ$, we derive estimates for the energy of initial data localized in $\mathscr{O}$, in terms of the energy measured on the observation set $(0,T) \times ω$. This holds under a suitable geometric condition relating the time horizon T and the subdomains $ω$ and $\mathscr{O}$. By duality, we obtain new controllability results for the wave equation, ensuring that the projection of the solution onto $\mathscr{O}$ can be controlled by means of controls supported in $ω$, with optimal spatial support. We also present several extensions of the main result, including the case of boundary observations, as well as a characterization of the observable fraction of the energy of the initial data from partial measurements on $(0,T) \times ω$. Applications to wave control are discussed accordingly. |
| title | Regional and partial observability and control of waves |
| topic | Analysis of PDEs 35A18, 35Lxx, 35Q93, 93Dxx |
| url | https://arxiv.org/abs/2504.18976 |