10/10/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Marius Tucsnak, Institut Élie Cartan - Université de Nancy
Fast and strongly localized observation for the Schrödingerequation in a rectangular domain
We provide necessary and sufficient for the exact observability of systems governed by the Schrödinger equation in a rectangle with Dirichlet or Neumann boundary conditions. Generalizing results from a previuos work with Ramdani, Takahashi and Tenenbaum, we prove that the corresponding criterion is that the observation region has non empty interior in the case of Dirichlet observation and, in the case of Neumann observation, that it has an open intersection with an edge of each direction. Thus, in both circumstances, observation regions may have arbitrarily small measures. We complement these results by proving that the above mentioned properties hold in arbitrarily small time. We also show that similar results hold for the Euler-Bernoulli plate equation. Finally, we give explicit estimates for the blow-up rate of the observability constants as the time and (or) the size of the observation region tend to zero. The main ingredients of the proofs are an effective version of a theorem of Beurling on non harmonic Fourier series and an estimate for the number of lattice points in a neighbourhood of an ellipse.
This is a joint work with G. Tenenbaum (Institut Élie Cartan).
