Subelliptic wave equations are never observable

Letrouit, Cyril

doi:10.2140/apde.2023.16.643

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN: 1948-206X (e-only)

ISSN: 2157-5045 (print)

Author index

To appear

Other MSP journals

Subelliptic wave equations are never observable

Cyril Letrouit

Vol. 16 (2023), No. 3, 643–678

DOI: 10.2140/apde.2023.16.643

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, the GCC is never satisfied, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $Δ = - \sum_{i = 1}^{m} X_{i}^{*} X_{i}$ on a manifold $M$ , and any measurable subset $ω \subset M$ such that $M ∖ ω$ contains in its interior a point $q$ with $[X_{i}, X_{j}] (q) \notin Span (X_{1}, \dots, X_{m})$ for some $1 \leq i, j \leq m$ , we show that, for any $T_{0} > 0$ , the wave equation with subelliptic Laplacian $Δ$ is not observable on $ω$ 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 $M ∖ ω$ . 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

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 16, No. 3, 2023