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 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.     

Contrôlabilité exacte des approximations de Galerkin des équations de Navier-Stokes

We consider Navier-Stokes equations in a smooth domain of ℝⁿ, n = 2, 3, with Dirichlet boundary conditions. We introduce the classical Galerkin approximation and study its exact controllability when the control acts on an open non-empty subset of the domain. Under suitable assumptions on the elements of the Galerkin basis, we prove that this finite-dimensional system is exactly controllable. The proof combines HUM together with a classical fixed point argument, and relies on the cancellation properties of the non-linearity appearing in the Navier-Stokes equations.     

Un problème d'optimisation de forme pour la contrôlabilité exacte de l'équation des ondes 2D

We consider the wave equation defined on $\Omega ?{\mathbb{R}}^2$ and $\omega ?\Omega$. We designate by $v_{\omega }$ the distributed control of minimal $L^2(\omega \times (0,T))$ norm obtained with the Hilbert Uniqueness Method which stabilizes the system at time $T>0$. This Note addresses the question of the optimal position of ω in order to minimize $J:\omega \to 6v_{\omega }{6}_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^{1,1}(\Omega )$, we express the shape derivative of J as a curvilinear integral on ?ω (independently of any adjoint solution) leading to a descent algorithm. A numerical application is given. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Analyse spectrale et singularités d'un problème de transmission non coercif

This Note is devoted to the spectral analysis of an unbounded operator associated with a non-coercive transmission problem. Using an integral equation method, we show that, if the interface is regular, this operator is selfadjoint and has compact resolvent. If the interface has a corner, the study of the singularities using Mellin transform allows us to derive a necessary and sufficient condition on the contrast between the two media for selfadjointness. If the operator is not selfadjoint, a characterization of its selfadjoint extensions is given.     

Existence et unicité de solutions d'un problème de couplage fluide-structure bidimensionnel stationnaire

We study the well-posedness of a steady-state problem: we consider a two-dimensional viscous incompressible flow, which is modeled by the Stokes equations. The structure is monodimensional and the equations describing the behaviour of elastic medium are beam equations. The fluid domain is defined by the shape of the structure function, itself resulting from a stress distribution coming from the fluid. The problem is thus nonlinear and the equations we deal with are coupled. We prove its solvability through Schauder's fixed point theorem.     

Existence de solutions d'un problème de couplage fluide-structure bidimensionnel instationnaire

We study the well-posedness of a steady stale problem: we consider a two-dimensional viscous incompressible flow, which is modeled bv the Navier-Stokes equations. The structure is a rigid moving disc. The fluid domain depends on time and is defined by the position of the structure, itself resulting from a stress distribution coming from the fluid. The problem is then nonlinear and the equations we deal with are coupled. We prove its local solvability in time though two fixed point procedures.     

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.     

Perturbation singulière d'un problème de frottement sec non monotone

In this Note we deal with a singularly perturbed system constituted by a differential inclusion which has a unique solution for each value of the perturbation parameter. The associated degenerated problem, that corresponds to a dynamic dry friction problem, has many solutions. We show that perturbed problem solutions converge to a particular solution of the degenerated problem when the perturbation parameter goes to zero. The singular perturbation approach allows an analysis of a criterion used to select a solution of the degenerated problem, and suggests a method to study more elaborated dry friction problems.     

Régularité des solutions d'un problème variationnel sous contrainte de convexité

We study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [4]. We then extend this result to a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [5].     

Homogénéisation d'un problème de capillarité

We study the behavior of the height of a liquid contained in a vertical cylindrical tube where are immerged a periodic distribution (period ɛ) of cylindrical fibers with radius r < ɛ and whose material-liquid coefficient is λ. It is showed that the homogenized problem depends on the limit of λr/ɛ² as (ɛ, λ, r) → (0,0, 0). The second problem is when the border is made of two alternate periodic bands, the first of length r being incrusted in the second of length ɛ − r. Here, the homogenized problem depends on the limit of r/ɛ as (, r) → (0, 0). For this, we use epi-convergence techniques.     

Sur un type de problème singulier au sein d'un groupement

Résumé Origine épistémologique de cette étude: on envisage, dans un groupement, des problèmes infiniment voisins d'un problème impossible. A ce dernier titre, est retenue la recherche de surfaces vérifiant · dM=0 pour V non nul. Intérêt du guide offert en ce cas par deux règles heuristigues, ayant la constructivité, la stabilité, pour object respectifs. Usage, plus efficace, de divers recours à des groupements de problèmes. Je veux, avant toute chose, rendre hommage au professeur Enrico Bompiani pour son oeuvre mathématique ample et belle. Je m'excuse de m'en écarter beaucoup ici, en reprenant une étude qui me tenait en échec depuis 1937. Je commence par un cas un peu simplifié, livrant un relais vers le cas général. Je terminerai par des remarques relatives aux règles heuristiques, grace auxquelles j'ai finalement dominé le cas général. à M. Enrico Bompiani pour son Jubilé scientifique.     

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.     

Modélisation d'un problème de diffusion avec pyrolyse et ablation

We discuss existence and uniqueness for the solutions of an unusual parabolic quasilinear problem, coming from a heat conduction model problem (atmosphere re-entry of a body) with pyrolysis and weak ablation.     

Sur un problème de Steiner

Sans résumé     

Contrôlabilité de l'équation des ondes à densité rapidement oscillante à une dimension d'espace

We consider the one-dimensional wave equation with periodic density of period ε → 0. By a counterexample due to Avellaneda, Bardos, and Rauch, we know that the boundary controllability property does not hold uniformly as ε → 0. We prove that the control remains uniformly bounded if we control the projection of the solution over the subspace generated by the eigenfunctions associated with the eigenvalues λ ≤ C_ε⁻², C > 0 being small enough. This result is sharp in the sense that the control diverges when the projection over the eigenfunctions such that λ ~ C_ε⁻², with C large, is controlled. We use the classical WKB asymptotic development that provides sharp results on the convergence of the spectrum and the theory of non-harmonic Fourier series.     

Temps de vie des solutions d'un problème de Cauchy non linéaire

We solve the global Cauchy problem, with small initial data, in the space of the holomorphic functions with respect to t and Gevrey class with respect to x. We establish the existence and the stability of the solution to Cauchy problem with nul initial data without hyperbolicity hypothesis. In the stationary case, we give estimates of life span of the solutions with respect to size of the initial data.     

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]).