Geometric control condition for the wave equation with a time-dependent observation domain
Investigated is the observability of the classical wave equation \(\square_{g}u=0\), on a \(d\)-dimensional Riemannian manifold \((\Omega,g)\), with homogeneous Dirichlet or Neumann boundary conditions. It is known that the observability property holds if some geometric control conditions, GCC are satisfied. The aim of the article is to extend the GCC to time-dependent observation domains. By using the notion of compressed generalized bicharacteristic, the time-dependent geometric control condition, t-GCC is defined. There is proved that under some special assumptions and if \(Q\) is an open subset of \((\mathbb R \times\overline{\Omega})\) that satisfies the t-GCC and \(T>T_{0}(Q,\Omega)\), the observability inequality holds. Further, as consequences, a result for the control of the wave equation with time-dependent control domain and results on observability with few sensors and stabilization are shown. The main results are then extended to the case of boundary observability. An extensive discussion is dedicated to examples and counterexamples.
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