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Subelliptic wave
equations are never observable
Cyril Letrouit |
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Vol. 16 (2023), No. 3, 643–678
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Abstract |
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It is well known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time is almost equivalent to the geometric control condition (GCC), which stipulates that any geodesic ray meets the control set within time . We show that in the subelliptic setting, the GCC is never satisfied, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian on a manifold , and any measurable subset such that contains in its interior a point with for some , we show that, for any , the wave equation with subelliptic Laplacian is not observable on in time . 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 . 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. |
Keywords
subelliptic, wave equation, observability, sub-Riemannian
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Mathematical Subject Classification 2010
Primary: 22E25, 35H20, 35L05, 93B05, 93B07
Secondary: 35H10, 35S05, 78A05
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Milestones
Received: 3 February 2020
Revised: 4 September 2021
Accepted: 6 October 2021
Published: 25 May 2023
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Authors |
| Cyril Letrouit | |
| Sorbonne Université, Université
Paris-Diderot, CNRS, Inria, Laboratoire Jacques-Louis Lions,
Paris France |
Department of Mathematics and
Applications École Normale Supérieure, CNRS, PSL Research University Paris France |
| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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