Cours-Udine-Puel-Mod1-06-04

Global Carleman inequalities for the waveequations and applications to controllabilityand inverse problems1Jean-Pierre Puel1J.-P. Puel: (jppuel@cmapx.polytechnique.fr) Laboratoire de Math´ematiques Ap-pliqu´ees, Universit´e de Versailles Saint-Quentin, 45 avenue des Etats Unis, 78035 VersaillesChapter 1Presentation of the problemsExact controllability problems for wave equations have been extensively studied inthe last fifteen years (see [13] for example) and this problem corresponds to manyapplications like how tostop vibrationsbyaboundaryaction forexample andmanyothers. We will consider here the case of wave equations with bounded potentialsand present a direct approach to this problem. Another problem which will be con-sidered here is to retrieve an unknown potential in a wave equation from boundarymeasurements. This is now an inverse problem and again it corresponds to impor-tant applications, even if a moreinteresting problem would beto retrieve a diffusioncoefficient or elasticity coefficients. The situation considered here constitutes onestep in this direction.The results we present here are not new but they are given in a unified way andthe attempt is to make them rather simple and comprehensible despite of some longcomputations.The link between the two questions appears clearly from Lions’ Hilbert Unique-ness Method for example. Here we obtain results on the two problems using thesame mathematical machinery.We will present this machinery which ...