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 $\Delta =-{\sum }_{i=1}^m{X}_i^{\ast }{X}_i$ on a manifold $M$, and any measurable subset $\omega \subset M$ such that $M\backslash \omega$ contains in its interior a point $q$ with $[X_i,X_j](q)\notin \text{Span}(X_1,\dots ,X_m)$ for some $1\le i,j\le m$, 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.

arXiv commentAnalysis & PDE, Mathematical Sciences Publishers

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Cited by in corpus (3)

• Time optimal observability for Grushin Schrödinger equation
• Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates
• Exact observability properties of subelliptic wave and Schr{ö}dinger equations