Geometric and probabilistic results for the observability of the wave equation
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 431-461.

Given any measurable subset ω of a closed Riemannian manifold and given any T>0, we define T (ω)[0,1] as the smallest average time over [0,T] spent by all geodesic rays in ω. Our first main result, which is of geometric nature, states that, under regularity assumptions, 1/2 is the maximal possible discrepancy of T when taking the closure. Our second main result is of probabilistic nature: considering a regular checkerboard on the flat two-dimensional torus made of n 2 square white cells, constructing random subsets ω ε n by darkening cells randomly with a probability ε, we prove that the random law T (ω ε n ) converges in probability to ε as n+. We discuss the consequences in terms of observability of the wave equation.

Étant donné un sous-ensemble mesurable ω d’une variété riemannienne compacte et étant donné T>0, on définit T (ω)[0,1] comme étant le plus petit temps moyen passé par les rayons géodésiques dans ω. Notre premier résultat principal, qui est de nature géométrique, établit que, sous des conditions de régularité, T peut augmenter au maximum de 1/2 en passant à l’adhérence. Notre second résultat principal est de nature probabiliste : considérant un damier régulier sur le tore plat de dimension 2 formé de n 2 carrés blancs, construisant des ensembles aléatoires ω ε n en noircissant les carrés de manière aléatoire avec probabilité ε, on montre que la variable aléatoire T (ω ε n ) converge en probabilité vers ε lorsque n+. Nous discutons les conséquences en termes d’observabilité de l’équation des ondes.

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DOI: 10.5802/jep.186
Classification: 93B07, 53C22, 60B10
Keywords: Observability, wave equation, Riemannian geometry, random set
Mot clés : Observabilité, équation des ondes, géométrie riemannienne, ensemble aléatoire

Emmanuel Humbert 1; Yannick Privat 2; Emmanuel Trélat 3

1 Institut Denis Poisson, UFR Sciences et Technologie, Faculté François Rabelais Parc de Grandmont, 37200 Tours, France
2 IRMA, Université de Strasbourg, CNRS UMR 7501 7 rue René Descartes, 67084 Strasbourg, France & Institut Universitaire de France (IUF)
3 Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions (LJLL) F-75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Emmanuel Humbert; Yannick Privat; Emmanuel Trélat. Geometric and probabilistic results for the observability of the wave equation. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 431-461. doi : 10.5802/jep.186. https://jep.centre-mersenne.org/articles/10.5802/jep.186/

[1] C. Bardos, G. Lebeau & J. Rauch - “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30 (1992) no. 5, p. 1024-1065 | DOI | MR | Zbl

[2] M. Berger - A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003 | DOI

[3] N. Burq - “Second microlocalization and stabilization of damped wave equations on tori”, in Shocks, singularities and oscillations in nonlinear optics and fluid mechanics, Springer INdAM Ser., vol. 17, Springer, Cham, 2017, p. 55-73 | DOI | MR | Zbl

[4] N. Burq & P. Gérard - “Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes”, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) no. 7, p. 749-752 | DOI | Zbl

[5] N. Burq & P. Gérard - “Stabilization of wave equations on the torus with rough dampings”, Pure Appl. Anal. 2 (2020) no. 3, p. 627-658 | DOI | MR | Zbl

[6] P. Hébrard & E. Humbert - “The geometrical quantity in damped wave equations on a square”, ESAIM Control Optim. Calc. Var. 12 (2006) no. 4, p. 636-661 | DOI | MR | Zbl

[7] E. Humbert, Y. Privat & E. Trélat - “Observability properties of the homogeneous wave equation on a closed manifold”, Comm. Partial Differential Equations 44 (2019) no. 9, p. 749-772 | DOI | MR | Zbl

[8] G. Lebeau - “Control for hyperbolic equations”, in Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992), École Polytechnique, Palaiseau, 1992, p. 24 | MR | Zbl

[9] J. Rauch & M. Taylor - “Exponential decay of solutions to hyperbolic equations in bounded domains”, Indiana Univ. Math. J. 24 (1974), p. 79-86 | DOI | MR

[10] M. Zworski - Semiclassical analysis, Graduate Studies in Math., vol. 138, American Mathematical Society, Providence, RI, 2012 | DOI

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