Gilles Lebeau

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A. Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates(More)
We consider the problem of the numerical approximation of the linear controllability of waves. All our experiments are done in a bounded domain Ω of the plane, with Dirichlet boundary conditions and internal control. We use a Galerkin approximation of the optimal control operator of the continuous model, based on the spectral theory of the Laplace operator(More)
1 Introduction Consider in the plane the flow of two ideal incompressible fluids of constant densities ρ ± > 0 in the gravity field, with ρ + = ρ −. Velocity field satisfies the Euler equation ∂ρu ∂t + u · ∇ρu = −∇p + ρg (1) div u = 0 (2) ρ t + div (ρu) = 0 (3) with initial data u(x, 0) = u 0 (x). (4) We suppose, that u 0 (x) satisfies the continuity(More)
A new Carleman inequality for parabolic systems with a single observation and applications Une nouvelle inégalité de Carleman pour des systèmes paraboliques avec une seule observation et applications a r t i c l e i n f o a b s t r a c t In this Note, we present Carleman estimates for linear reaction–diffusion–convection systems of two equations and linear(More)