Contrôlabilité exacte frontière d’un système hybride en élasticité

We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, we prove that the hybrid system is exactly controllable in an arbitrarily short time in the usual energy space.     

Stabilisation frontière d'un système d'élasticité

We prove in this Note the global existence, uniqueness and stabilization of an elasticity system with nonlinear boundary damping. This work extends results obtained by Alabau and Komornik in [1]     

Homogénéisation de frontière en élasticité linéaire

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of R³, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ɛ distributed on some part of the boundary of this domain, when the parameter ɛ tends to 0. We use epi-convergence arguments in order to establish this asymptotic behaviour.     

Contrôlabilité exacte d'un problème avec conditions de Ventcel évolutives pour le système linéaire de l'élasticité

We examine the exact controllability of the solution of the linear elasticity system with evolutive Ventcel conditions in a bounded domain of ø³. We use the Hubert uniqueness method (HUM) of Lions [7]; some multipliers are defined on the boundary: the curvature tensor [3] appears when computing some boundary integrals.     

Contrôlabilité asymptotique et synchronisation asymptotique dʼun système couplé dʼéquations des ondes avec des contrôles frontières de Dirichlet

In this Note, the asymptotic null controllability and various kinds of asymptotic synchronization, considered as some kinds of weakened controllability and synchronization, are introduced and studied for a coupled system of wave equations with Dirichlet boundary controls. Equivalent properties of weak observability are established.     

Observabilité et contrôlabilité exacte indirecte interne par un contrôle localement distribué de systèmes d'équations couplées

We study the observability and the exact controllability of a weakly coupled system with an internal locally control acting on only one equation. We show that, for sufficiently large time, the observation of the velocity of the first component of the solution on a neighborhood of a part of the boundary allows us to get back a weakened energy of initial data of the second component, this if the coupling parameter is sufficiently small, but non-vanishing. This result leads to a uniqueness theorem and, by the HUM method, we prove that the total system is exactly controllable.     

Stabilisation frontière indirecte du système de Timoshenko

In this Note, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. Under the equal speed wave propagation condition, we establish the exponential stability of the system. On the contrary, we show that the decay rate is polynomial.     

Synchronisation exacte dʼun système couplé dʼéquations des ondes par des contrôles frontières de Dirichlet

In this Note, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.     

Sur l'unicité pour le système de Boussinesq avec diffusion non linéaire

In this article we prove global existence and uniqueness result for the 2D Boussinesq system with non linear thermal diffusion in some critical spaces.     

Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces

We study the convergence of the solution u^ɛ of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder, the diameter of which tends to zero. We prove in particular that u^ɛ − (u + ɛv + ɛ²w) strongly converges to zero (in a sense which will be specified), where (u, v, w) is the unique solution of an elliptic system of partial differential equations.     

Stabilisation d'un système hybride par un feedback non linéaire,non monotone

For a hybrid system composed of a cable with masses at both ends, we prove strong stability for a nonlinear and nonmonotone feedback law applied at one end.     

Comportement asymptotique des solutions d'un système elliptique conservatif non linéaire

Let M be a compact riemannian manifold; in a previous article we show that every non- negative solution of u_tt + Δ_g u=f(u) on M ×R ⁺, satisfying Dirichlet or Neumann boundary conditions, converges to a (stationary) solution Φ of Δ_g Φ=f(Φ) with exponential decay of ∥u - Φ∥c²(M), if we assume that f behaves like r? r^p - λr. We extend this result to a system in the following form $$\left\{ {\begin{array}{*{20}c} {u_{tt} + \Delta _g u + \alpha u - G_x (u,\upsilon ) = 0,} \\ {u_{tt} + \Delta _g \upsilon + \beta \upsilon - G_x (u,\upsilon ) = 0,} \\ \end{array} } \right.$$ . where G satisfies some growth and convexity properties.     

Sur une équation différentielle linéaire du troisième ordre

Sans résuméExtrait d'une lettre adressée à Mr.F. Klein.     

Caractère bien posé du problème de Cauchy pour le système de Bérenger

The Bérenger perfectly matched layer is used in computational electromagnetism as an absorbing layer in scattering problems. It raises delicate mathematical issues. In this Note we show, for regular data, the existence and uniqueness of strong solutions to the Cauchy problem derived from the PML method. The result is presented in the 2-D case. The key to the proof is an appropriate control of a mixed H¹- L² norm of the solution by the same norm of the initial data. Beside a paper is in preparation about extensions of this results (L² estimates, 3-D case) (see also [5]).     

Solutions périodiques d'un système différentiel non linéaire du second ordre avec changement de signe

Résumé On s'intéresse à la recherche de solutions T périodiques d'un système + a(t)V(x)= =f(t), où a:R R et f:R Rⁿ sont T périodiques. Le potentiel V est supposé convexe sous quadratique et on utilise la formulation variationnelle duale du problème. On démontre différents résultats d'existence d'au moins une solution au problème, non triviale dans le cas où la fonction f est nulle.     

Contrôlabilité exacte d'un problème de transmission

By using HUM, we prove the exact controllability for the transmission wave equation in the case a₁<a₂.     

Approximation d'un problème de l'élasticité non linéaire par une méthode d'éléments finis mixtes

In this Note we introduce a new three field mixed formulation to approximate the solution of a nonlinear elasticity problem based on the Raviart–Thomas type elements. The existence and uniqueness of solutions of the mixed formulation and its discretization are proved and optimal error estimates are derived.