We characterize the observability property (and, by duality, the controllability
and the stabilization) of the wave equation on a Riemannian manifold
, with
or without boundary, where the observation (or control) domain is time-varying. We
provide a condition ensuring observability, in terms of propagating bicharacteristics.
This condition extends the well-known geometric control condition established for
fixed observation domains.
As one of the consequences, we prove that it is always possible to find a
time-dependent observation domain of arbitrarily small measure for which the
observability property holds. From a practical point of view, this means that it is
possible to reconstruct the solutions of the wave equation with only few sensors (in
the Lebesgue measure sense), at the price of moving the sensors in the domain in an
adequate way.
We provide several illustrating examples, in which the observation domain is the rigid displacement
in
of a fixed
domain, with speed
,
showing that the observability property depends both on
and
on the wave speed. Despite the apparent simplicity of some of our examples, the
observability property can depend on nontrivial arithmetic considerations.
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Université Paris-Nord, CNRS UMR
7339
Laboratoire d’analyse géométrie et applications (LAGA)
Institut universitaire de France
99 Avenue Jean Baptiste Clément
93430 Villetaneuse France