Stabilisation frontière indirecte de systèmes faiblement couplés

This work is concerned with the boundary stabilization of coupled equations where only one of the equation is stabilized (indirect damping, see e.g. Russell [9]). We show for different examples (wave equations, Kirchhoff plates) that the full system can be strongly stabilized by that way, provided that the coupling parameter is sufficiently small (but not equal to zero). In such cases we prove that the energy decays polynomially with explicit polynomial decay rates for sufficiently smooth solutions. We extend these results to the case of two coupled wave equations with different speeds of propagation under a condition of the ratio of the two speeds and for n-dimensional intervals.     

Sur les variétés à courbure positive de diamètre minimum

Sans résumé Le résultat de cet article a été présenté au Colloque international de Géométrie différentielle et de Topologie de Zurich de juin 1960.     

Sur la détermination du périmètre de l'ovale à huit centres

This note aims to present the history of the main researches carried out to represent an ellipse by means of an eight-centered oval and to complete them by giving a geometrical demonstration of the formulas expressing the coordinates of the centers, the radii and the central angles of these arcs of circles making up this oval. These formulas will make it possible to determine its perimeter and thus to approximate the value of that of the ellipse. This result could be useful for archaeologists, who will test the hypothesis according to which some Roman amphitheatres, such as the Coliseum, were built from eight-centered ovals.     

Contre-exemple à une caractérisation conjecturée de la sphère

It has long been conjectured that a closed convex surface of class C₊² whose principal curvatures K₁, K₂ satisfy the inequality (K₁−c)(K₂−c)≤0 with some constant c, must be a sphere. Partial results have been obtained by A.D. Aleksandrov, H.F. Münzner and D. Koutroufiotis.We reformulate the conjecture in terms of hedgehogs and we give a counter-example. Besides, we prove the conjecture for surfaces of constant width and give a new proof for analytic surfaces.     

Théorèmes du type Ingham et application à la théorie du contrôle

Ingham [4] improved a previous result of Wiener [10] on nonharmonic Fourier series. Modifying his weight function we obtain optimal results improving several earlier theorems of Kahane [7], Castro and Zuazua [2] and of Jaffard, Tucsnak, and Zuazua [5]. Then we apply these results to simultaneous observability problems.     

Sur la convergence numérique de différentes méthodes de décomposition pour une classe de problèmes elliptiques

Summary The present paper deals with some methods of decomposition where the numerical convergence for non linear differential equations is studied in the case of adapted decompositions of the right hand side.     

Problème inverse de la théorie du potentiel logarithmique pour les lemniscates

The paper is related to the question of uniqueness in the inverse logarithmic potential problem. This question is to find the conditions on which two domains D ₁ and D ₂ producing the same external potential must coincide. Assuming the general hypothesis of regularity and an additional condition of connectivity of (D₁D₂)_c, we prove a theorem of uniqueness in the case when one of the domains is a lemniscate. The main tool is one lemma for Cauchy's potential due to M. Sakai. We give a simple proof of its extension to Newtonian potential.

     

Solutions du système d'équations différentielles de Hua et intégrales de Poisson-Shilov des fonctions de carré intégrable sur la frontière de Shilov de la boule matricielle de rang deux

We give a characterization of those solutions F of the Hua system of differential equations H_DF(Z) = − (λ² + 4)F(Z) · I, that are Poisson-Shilov integrals of L²-functions on the Shilov boundary of the rank two matrix ball.     

Problème de Cauchy pour une équation parabolique modélisant la relaxation des systèmes stellaires auto-gravitants

In this Note we are concerned with the Cauchy problem for a parabolic equation (with nonlocal terms), which has recently been proposed by Chavanis, Robert and Sommeria as a model for the large-scale dynamics of stellar systems. We show the local existence and the uniqueness of a variational solution, together with the increasing of the entropy and the positiveness of the stellar density and of the temperature when the initial state is radial.     

Modèles de verre de spins à champ moyen et à interactions multiples : résultats rigoureux à basse température

We study the Ising case of the p-spin interaction mean field model for spin glasses. For large p, below the critical temperature, the configuration space naturally decomposes in a sequence of “states”. In the thermodynamic limit, the overlaps between two generic elements of any two different “states” are zero.We also construct systems that provably exhibit several phase transitions between what one might in replica theory call different levels of symmetry breaking.     

Un résultat de stabilité pour la récupération d'un paramètre du système de la viscoélasticité 3D

In this Note, we prove a Carleman's estimate for the integro-differential hyperbolic system of the viscoelasticity problem and we use this estimate to obtain a stability result for the inverse problem of recovering a viscoelastic coefficient from a unique internal measure. To cite this article: M. de Buhan, A. Osses, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Estimation de la matrice de variance asymptotique des estimateurs du QMV de modèles ARMA faibles multivariés

In this Note, we consider the problems of estimating the asymptotic variance of the quasi-maximum likelihood estimator (QMLE) of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak VARMA). We first give expressions for the derivatives of the VARMA residuals in terms of the parameters of the models. Secondly we give an explicit expression of the asymptotic variance of the QMLE, in terms of the VAR and MA polynomials, and of the second- and fourth-order structure of the noise. We deduce a consistent estimator of the asymptotic variance of the QMLE.     

Bornes sur la densité pour un problème de Navier–Stokes compressible à frontière variable avec conditions aux limites de Dirichlet

We extend a compactness result shown by P.-L. Lions in 1998 to an isentropic compressible Navier–Stokes problem ($\gamma ?1$) defined on a time dependent domain with Dirichlet boundary conditions. This result can be useful for the study of some fluid–structure interaction problems, for the analysis of some pollution water problems (shallow water equations with free boundary: $\gamma =1$) or for the modelling of a river level. To cite this article: F. Flori, B. Giudicelli, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Sur la formation de microstructures et leur analyse numérique dans les problèmes à puits

We analyze the appearance of microstructures in the problems of Calculus of Variations with wells. We consider in particular the vector case with incompatible wells and we show how incompatibility gives rise to specific microstructures that we investigate from a numerical point of view.     

Problème de Helmholtz intérieur : formulation modale couplée à une représentation intégrale

Let Ω be an opened domain of ${\mathbb{R}}^3$ with a boundary Γ. The problem numbered (1) in the text has a unique solution in ${\mathrm{H}}^1(\Omega )$, if $k\in \mathbb{R}$, $R_e(\frac{1}{\zeta })>0$ on a part of Γ the area S of which is different from zero and $g(k,y,?)\in {\mathrm{H}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(\Gamma )$. ? is the damping of an elastic structure and ζ is the normalised acoustic impedance of the internal wall of the cavity. ? and $\frac{1}{\zeta }$ are small parameters. Thanks to a proper modal expansion and a mean over a narrow band of wave number k, an integral relation between the trace of u on $\Gamma ,\zeta ,?$ and g is built to the first order $?(\frac{1}{\zeta },?)$ in $\frac{1}{\zeta }$ and ?, using the residues theorem. It is not an equivalent equation to the problem, but just a step towards its resolution, which will be published in future papers. To cite this article: D. Brenot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).