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Subelliptic wave equations are never observable

Cyril Letrouit

Vol. 16 (2023), No. 3, 643–678
Abstract

It is well known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time T0 is almost equivalent to the geometric control condition (GCC), which stipulates that any geodesic ray meets the control set within time T0. 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 Δ = i=1mXiXi on a manifold M, and any measurable subset ω M such that Mω contains in its interior a point q with [Xi,Xj](q)Span (X1, ,Xm) for some 1 i,j m, we show that, for any T0 > 0, the wave equation with subelliptic Laplacian Δ is not observable on ω in time T0.

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 Mω. 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
Mathematical Subject Classification 2010
Primary: 22E25, 35H20, 35L05, 93B05, 93B07
Secondary: 35H10, 35S05, 78A05
Milestones
Received: 3 February 2020
Revised: 4 September 2021
Accepted: 6 October 2021
Published: 25 May 2023
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

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