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Subelliptic wave equations are never observable

Abstract : It is well-known that observability (and, by duality, controllability) 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 the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $\Delta=-\sum_{i=1}^m X_i^*X_i$ on a manifold $M$, and any measurable subset $\omega\subset M$ such that $M\backslash \omega$ contains in its interior a point $q$ with $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we show that for any $T_0>0$, the wave equation with subelliptic Laplacian $\Delta$ is not observable on $\omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the associated sub-Riemannian distance) spending a long time in $M\backslash \omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.
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https://hal.archives-ouvertes.fr/hal-02466229
Contributor : Cyril Letrouit Connect in order to contact the contributor
Submitted on : Monday, October 11, 2021 - 6:50:29 PM
Last modification on : Friday, December 3, 2021 - 11:43:52 AM

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  • HAL Id : hal-02466229, version 5
  • ARXIV : 2002.01259

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Cyril Letrouit. Subelliptic wave equations are never observable. Analysis & PDE, Mathematical Sciences Publishers, In press. ⟨hal-02466229v5⟩

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