In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in R 3 with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p < 5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cutoff away from the boundary.  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous étudions dans cet article la décroissance exponentielle de l'énergie pour une équation d'ondes semi-linéaire dans R 3 , avec un terme d'amortissement effectif à l'extérieur d'une boule. En supposant la non linéarité sous critique et vérifiant certaines conditions naturelles, nous obtenons un résultat de stabilisation locale, c'est-à-dire une décroissance exponentielle de l'énergie, uniforme sur les boules de l'espace d'énergie où sont choisies les données initiales. La démonstration repose sur des inégalités d'énergie classiques qui estiment l'énergie totale en fonction de l'énergie localisée à l'extérieur d'une boule. Elle utilise aussi les estimations de Strichartz et les résultats de P. Gérard sur les mesures de défaut microlocales et les suites linéarisables. Nous donnons aussi, en application, un résultat de stabilisation et de contrôle pour l'équation des ondes semi-linéaire sur un ouvert borné, avec une non linéarité sous critique, tronquée loin du bord.  2003 Éditions scientifiques et médicales Elsevier SAS 1 Supported by grant BFM2002-03345 of the MCYT (Spain) and the Networks "Homogeneization and Multiple Scales" and "New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulation (HPRN-CT-2002-00284)" of the EU.

In this paper, we study the stabilization property and the exact controllability for the nonlinear Schrödinger equation on a two dimensional compact Riemannian manifold, without boundary. We use a pseudo-differential dissipation. The proofs are based on a result of propagation of singularities and on recent dispersion estimates (Strichartz type inequalities) due to N. Burq, P. Gérard and N. Tzvetkov.

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We consider the exact controllability problem on a compact manifold Ω for two coupled wave equations, with a control function acting on one of them only. Action on the second wave equation is obtained through a coupling term. First, when the two waves propagate with the same speed, we introduce the time Tω→O→ω for which all geodesics traveling in Ω go through the control region ω, then through the coupling region O, and finally come back in ω. We prove that the system is controllable if and only if both ω and O satisfy the Geometric Control Condition and the control time is larger than Tω→O→ω. Second, we prove that the associated HUM control operator is a pseudodifferential operator and we exhibit its principal symbol. Finally, if the two waves propagate with different speeds, we give sharp sufficient controllability conditions on the functional spaces, the geometry of the sets ω and O, and the minimal time.