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 e quation may be obtained as limit of the exact controllability properties of singularly perturbed damped wave equations, We impose Dirichlet, homogeneo us boundary conditions. The control is supported in a neighborhood of a sub set of the boundary that satisfies the classical requirements to apply mult iplier techniques. The proof needs an iterative argument that allows to tre at separately the low and high frequencies and to make use of the dissipati vity of the systems under consideration. This is combined with sharp observ ability estimates on the eigenfunctions of the Laplacian due to G. Lebeau a nd L. Robbiano, and global Carleman estimates. This proof applies in any space dimension. As a consequence of the uniform controllability we derive uniform observabi lity estimates which can not be proved by classical methods due to the sing ular character of the perturbations we deal with. (C) 2000 Editions scienti fiques et medicales Elsevier SAS.