Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations

In this paper we prove that the null controllability property of the heat equation may be obtained as limit of the exact controllability properties of singularly perturbed damped wave equations, We impose Dirichlet, homogeneous boundary conditions. The control is supported in a neighborhood of a subset of the boundary that satisfies the classical requirements to apply multiplier techniques. The proof needs an iterative argument that allows to treat separately the low and high frequencies and to make use of the dissipativity of the systems under consideration. This is combined with sharp observability estimates on the eigenfunctions of the Laplacian due to G. Lebeau and L. Robbiano, and global Carleman estimates. This proof applies in any space dimension. As a consequence of the uniform controllability we derive uniform observability estimates which can not be proved by classical methods due to the singular character of the perturbations we deal with. (C) 2000 Editions scientifiques et medicales Elsevier SAS.