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Journal Articles Journal de l'École polytechnique — Mathématiques Year : 2022

Geometric and probabilistic results for the observability of the wave equation

Abstract

Given any measurable subset $\omega$ of a closed Riemannian manifold and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. Our first main result, which is of geometric nature, states that, under regularity assumptions, $1/2$ is the maximal possible discrepancy of $\ell^T$ 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 $\omega_\varepsilon^n$ by darkening cells randomly with a probability $\varepsilon$, we prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$. We discuss the consequences in terms of observability of the wave equation.
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Dates and versions

hal-01652890 , version 1 (30-11-2017)
hal-01652890 , version 2 (05-12-2019)
hal-01652890 , version 3 (16-02-2022)

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Cite

Emmanuel Humbert, Yannick Privat, Emmanuel Trélat. Geometric and probabilistic results for the observability of the wave equation. Journal de l'École polytechnique — Mathématiques, 2022, Tome 9, pp.431--461. ⟨10.5802/jep.186⟩. ⟨hal-01652890v3⟩
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