We characterize the observability property (and, by duality, the
controllability and the stabilization) of the wave equation on a Riemannian
manifold Ω, 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 observationdomain is the rigid displacement in Ω of a fixed
domain, withspeed v, showing that the observability property depends both on
vand on the wave speed. Despite the apparent simplicity of some of
ourexamples, the observability property can depend on nontrivial
arithmeticconsiderations