Observability properties of the homogeneous wave equation on a closed manifold
| Autor: | Emmanuel Humbert, Emmanuel Trélat, Yannick Privat |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université de Tours-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, Centre National de la Recherche Scientifique (CNRS)-Université de Tours |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
Closed manifold Geodesic [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] 01 natural sciences Mathematics - Analysis of PDEs FOS: Mathematics Geometric control condition [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Observability 0101 mathematics Mathematics - Optimization and Control Mathematical physics Mathematics Observability inequality Applied Mathematics 010102 general mathematics Eigenfunction Riemannian manifold Wave equation Infimum and supremum Functional Analysis (math.FA) Mathematics - Functional Analysis 010101 applied mathematics Differential Geometry (math.DG) Optimization and Control (math.OC) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] Laplace operator Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Communications in Partial Differential Equations Communications in Partial Differential Equations, 2019, 44 (9), pp.749--772. ⟨10.1080/03605302.2019.1581799⟩ Communications in Partial Differential Equations, Taylor & Francis, 2019, 44 (9), pp.749--772. ⟨10.1080/03605302.2019.1581799⟩ |
| ISSN: | 0360-5302 1532-4133 |
| Popis: | International audience; We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a measurable subset $\omega$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $\omega$ is open and if the pair $(\omega,T)$ satisfies the Geometric Control Condition then an observability inequality is satisfied, comparing the total energy of solutions to their energy localized in $\omega \times (0, T)$. The observability constant $C_T( {\omega})$ is then defined as the infimum over the set of all nontrivial solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In this paper, we provide estimates of the observability constant based on a low/high frequency splitting procedure allowing us to derive general geometric conditions guaranteeing that the wave equation is observable on a measurable subset $\omega$. We also establish that, as $T\rightarrow+\infty$, the ratio $C_T( {\omega})/T$ converges to the minimum of two quantities: the first one is of a spectral nature and involves the Laplacian eigenfunctions; the second one is of a geometric nature and involves the average time spent in $\omega$ by Riemannian geodesics. |
| Databáze: | OpenAIRE |
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