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Cyril Letrouit

Sub-Riemannian wave equations are never observable
Jeudi, 23 Janvier, 2020 - 14:00
Résumé : 
In this talk, we explain a result we obtained recently, concerning the wave equation with a sub-Riemannian (i.e. hypoelliptic) Laplacian. Given a manifold $M$, a measurable subset $\omega\subset M$, a time $T_0$ and a sub-Riemannian Laplacian $\Delta$ on $M$, we say that the wave equation with Laplacian $\Delta$ is observable on $\omega$ in time $T_0$ if any solution $u$ of $\partial_{tt}^2u-\Delta u=0$ with fixed initial energy satisfies $\int_0^{T_0}\int_\omega |u|^2dxdt\geq C$ for some constant $C>0$ independent on $u$.
 
It is known since the work of Bardos-Lebeau-Rauch that  the observability of the elliptic wave equation, i.e. with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the  geometric control condition (GCC), which stipulates that any geodesic ray meets $\omega$ within time $T_0$. We show that in the sub-Riemannian setting, as soon as $M\backslash \omega$ has non-empty interior and $\Delta$ is ``sub-Riemannian but not Riemannian", GCC is never verified, which implies that sub-Riemannian wave equations are never observable. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the distance on $M$ associated to $\Delta$) spending a long time in $M\backslash \omega$.
Institution de l'orateur : 
Laboratoire Jacques-Louis Lions
Thème de recherche : 
Théorie spectrale et géométrie
Salle : 
4
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