# Subelliptic wave equations are never observable

2 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
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 $M$ such that $\text{Lie}(X_1,\ldots,X_m)=TM$ but $\text{Span}(X_1,\ldots,X_m)\subsetneq TM$, we show that for any $T_0>0$ and any measurable subset $\omega\subset M$ such that $M\backslash \omega$ has nonempty interior, 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 spiraling 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 <>
Submitted on : Thursday, November 19, 2020 - 11:10:12 AM
Last modification on : Tuesday, December 8, 2020 - 3:43:22 AM

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### Identifiers

• HAL Id : hal-02466229, version 2
• ARXIV : 2002.01259

### Citation

Cyril Letrouit. Subelliptic wave equations are never observable. 2020. ⟨hal-02466229v2⟩

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