Waves, damped wave and observation

This talk describes some applications of two kinds of observation estimate for the wave equation and for the damped wave equation in a bounded domain where the geometric control condition of C. Bardos, G. Lebeau and J. Rauch may failed. 1 The wave equation and observation We consider the wave equation in the solution u = u(x, t)    ∂ t u−∆u = 0 in Ω× R , u = 0 on ∂Ω× R , (u, ∂tu) (·, 0) = (u0, u1) , (1.1) living in a bounded open set Ω in R, n ≥ 1, either convex or C and connected, with boundary ∂Ω. It is well-known that for any initial data (u0, u1) ∈ H(Ω) ∩ H 0 (Ω) × H 0 (Ω), the above problem is well-posed and have a unique strong solution. Linked to exact controllability and strong stabilization for the wave equation (see [Li]), it appears the following observability problem which consists in proving the following estimate ‖(u0, u1)‖2H1 0 (Ω)×L2(Ω) ≤ C ∫ T