Observabilité, contrôlabilité et stabilisation frontière du système d’élasticité linéaire

We study homogeneous but not necessarily isotropic linear elastodynamic systems. Applying the Hilbert Uniqueness Method (HUM), we establish their boundary observability and their exact boundary controllability. Then we construct boundary feedbacks leading to arbitrarily large energy decay rates. Finally, under some geometrical conditions, we prove that a “natural” boundary feedback leads also to exponentiel energy decay.     

Une nouvelle famille de multiplicateurs et applications à la contrôlabilité exacte de l'équation d'ondes

Two new choices in the multiplier method to study the exact controllability of the wave equation are introduced: a new family of multipliers with rotated direction and multipliers with multiple directions. New geometrical examples with explicit estimates of the observability constant are found.     

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.     

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.     

Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D

We present a parameterized family of finite difference schemes for the exact controllability of the 1-D wave equation. These schemes differ from the usual centered ones by additional terms of order to $h^2$, where h denotes the discretization step in space. Using a discrete version of Ingham's inequality for nonharmonic Fourier series, the spectral properties and dispersion diagrams of the schemes, we determine the parameters leading to a uniform controllability property with respect to h and an optimal stability CFL condition. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Configurations combinatoires des polynômes fibrés de degré 2

In this Note, we are interested in quadratic fibered polynomials which are skew products of the kind P_c : X × C → X × C, P_c(x, z) = (ƒ(x),z² + c(x)), where X is a compact set, ƒ is a continuous map from X into X, and c is a continuous complex valued function on X considered as a parameter. We construct a compact connected configuration space which gives a combinatorial model of a subset of the parameters space. Then, we explain in which way an abstract configuration can be realized by a quadratic fibered polynomial.     

Clôture Intégrale D'idéaux et la Propriété (Z k )

Dans cet article nous donnons quelques applications du théoréme de valuations de Rees à l’étude de la clôture intégrale des idéaux. En particulier, nous étudions la question de Hübl et Swanson (Cf. [2 Hübl , R. , Swanson , I. ( 2001 ). Discrete valuations centered on local domains . J. Pure Appl. Algebra 161 ( 1–2 ): 145 – 166 .[Crossref], [Web of Science ®] , [Google Scholar]], Question 2.9). Ensuite, nous introduisons une nouvelle propriété des idéaux, que nous l'appelons (Z _k ). Cette propriété nous permettra de majorer le nombre de valuations de Rees associées à un idéal I.     

Une borne effective sur l'écart entre le polytope de contrôle et le graphe d'un polynôme réel sur un simplexe

We provide in this Note an explicit bound on the gap between the control polytope and the graph of a real polynomial, expressed in the simplicial Bernstein basis. This generalizes known results in dimensions 1 and 2. To cite this article: R. Leroy, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Décomposition de domaine pour le contrôle de systèmes gouvernés par des équations d'évolution

We present a non overlapping iterative domain decomposition method with “coupled” Robin transmission conditions. We prove its convergence on an optimal control problem for the wave equation. The linear part of the “feed-back” law associated to the local optimal control problems set on subdomains is independent of the iterative process. The method can be applied, at least formally, to the optimal control of systems governed by evolution equations.     

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.     

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.     

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).     

Normalité asymptotique de l'estimateur par polynômes locaux de la densité et ses dérivées pour des processus réels et mélangeants

We establish the convergence in probability and the asymptotic normality of the kernel polynomial estimators of the density function and its derivatives when the observed process is strictly stationary and strong mixing.     

Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

Variété de Picard d'une variété complète

Résumé On construit la variété dePicard d'une variété complète arbitraire U, c'est-à-dire, qu'on donne une structure canonique de variété de groupe au groupe des classes des diviseurs (localment principaux) de U algébriquement équivalents à zéro. à M. Enrico Bompiani pour son Jubilé scientifique.     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?