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.