We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.

In this paper we give a general presentation ofthe homogenization of Neumann type problems in periodically perforateddomains, including the case where the shape of the reference hole varies with the sizeof the period (in the spirit of the construction of self-similar fractals).We shows that H 0-convergence holds under the extra assumption thatthere exists a bounded sequence of extension operators forthe reference holes. The general classof Jones-domains gives an example where this result applies. When this assumption fails, another approach, usingthe Poincaré–Wirtingerinequality is presented. A corresponding class where it appliesis that of John-domains, for which the Poincaré–Wirtinger constantis controlled.The relationship between these two kinds of assumptions is also clarified.

We prove uniqueness of the invariant measure and the exponential convergence to equilibriumfor a stochastic dissipative system whose drift is perturbed by a bounded function.

We state and prove a Korn-like inequality for a vector field in abounded open set of $\mathbb{R}^N$ , satisfying a tangency boundary condition.This inequality, which is crucial in our study of the trend towardsequilibrium for dilute gases, holds true if and only if the domain is notaxisymmetric. We give quantitative, explicit estimates on how thedeparture from axisymmetry affects the constants; a Monge–Kantorovichminimization problem naturally arises in this process. Variants in the axisymmetric case are briefly discussed.

The results of this paper concern exact controllability to thetrajectories for a coupled system of semilinear heat equations. Wehave transmission conditions on the interface and Dirichlet boundaryconditions at the external part of the boundary so that the system can beviewed as a single equation with discontinuous coefficients in theprincipal part. Exact controllability to the trajectories is proved when weconsider distributed controls supported in the part of the domain where thediffusion coefficient is the smaller and if the nonlinear term f(y) growsslower than |y|log3/2(1+|y|) at infinity. In the proof we use nullcontrollability results for the associate linear system and globalCarleman estimates with explicit bounds or combinations of several ofthese estimates. In order to treat the terms appearing on theinterface, we have to construct specific weight functions depending ongeometry.

Le but de cet article est de montrer l'état de l'art dans l'analyse des systèmes distribués lorsque l'on examine certains concepts à partir de considérations régionales. Autrement dit, à partir de la donnée d'un système dynamique défini sur un domaine Ω, on ne s'intéresse à sa contrôlabilité, à son observabilité, à sa stabilité, ... que sur une région privilégiée ω, $\omega \subset \Omega$. Partant de concepts classiques, on développe leur adaptation au cas régional. On développe ensuite des concepts régionaux propres aux systèmes distribués tels que l'étalabilité. Divers développements spécifiques sont donnés. De nombreux résultats et exemples illustrent le travail.

We discuss some implications of linear programming for Mather theory [13-15] and itsfinite dimensional approximations. We find that the complementaryslackness condition of duality theory formally implies that the Mather set lies in ann-dimensional graph and as well predicts the relevant nonlinear PDE for the “weakKAM” theory of Fathi [5-8].

We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.

On expose les difficultés d'ordremathématique que posent des modèles récents de sédimentation-érosion de bassins élaborés par l'InstitutFrançais du Pétrole et fondés sur la prise en compte dediverses contraintes d'unilatéralité. On présente quelquesrésultats partiels théoriques et des directions de recherche pour larésolution d'un problème inverse posé par l'étudestratigraphique d'une colonne monolithologique.

The receding horizon control strategy fordynamical systems posed in infinite dimensional spaces is analysed. Itsstabilising property is verified provided controlLyapunov functionals are used as terminal penalty functions.For closed loop dissipative systems the terminal penalty canbe chosen as quadratic functional. Applications to the Navier–Stokesequations, semilinear wave equations and reaction diffusion systems are given.

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the K-method of real interpolation.

We consider a boundary optimal control problem for the Maxwell system with a final value cost criterion. We introduce a time domain decomposition procedure for the corresponding optimality system which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. In the limit full recovery of the coupling conditions is achieved, and, hence, the local solutions and controls converge to the global ones. The process is inherently parallel and is suitable for real-time control applications.

Nous prouvons que pour toute solution u du problème de Kelvin–Helmholtz des nappes de tourbillons pour l'équation d'Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de u et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de u sur t=0 lorsque u est définie sur un demi-interval [O,T[.

We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.

In this note, we propose a general definition of shape which isboth compatible with the one proposed in phenomenology(gestaltism) and with a computer vision implementation. We reversethe usual order in Computer Vision. We do not define “shaperecognition" as a task which requires a “model" pattern which issearched in all images of a certain kind. We give instead a“blind" definition of shapes relying only on invariance and repetition arguments.Given a set of images $\cal I$ , we call shape of this set anyspatial pattern which can be found at several locations of someimage, or in several different images of $\cal I$ . (This meansthat the shapes of a set of images are defined without any a priori assumption or knowledge.) The definition is powerful whenit is invariant and we prove that the following invariancerequirements can be matched in theory and in practice: localcontrast invariance, robustness to blur, noise and sampling,affine deformations. We display experiments with single images and image pairs. In eachcase,we display the detected shapes. Surprisingly enough, but in accordancewith Gestalt theory,the repetition of shapes is so frequent in human environment, that manyshapes can even be learnedfrom single images.

A class of variationaldata assimilation problems on reconstructingthe initial-value functions is considered for the models governed byquasilinear evolution equations. The optimality system is reduced to theequation for the control function. The properties of the control equation are studied and thesolvability theorems are proved for linear and quasilinear data assimilationproblems. The iterative algorithms for solving the problem are formulated andjustified.

We study the homogenization of the compressible Navier–Stokes system in a periodic porous medium (of period ε) with Dirichlet boundary conditions. At the limit, we recover different systems depending on the scaling we take. In particular, we rigorously derive the so-called “porous medium equation”.

We wish to show how the shock position in a nozzle could becontrolled. Optimal control theory and algorithm is applied to thetransonic equation. The difficulty is that the derivative withrespect to the shock position involves a Dirac mass. The onedimensional case is solved, the two dimensional one is analyzed .

This paper uses minimization methods and renormalized functionals to find spatially heteroclinic solutions for some classes of semilinear elliptic partial differential equations

Dans ce papier nous prouvons que si une solution de KdV est suffisammentdécroissante à l'infini (c'est-à-dire comme e $^{-x^\alpha} $ où $\alpha >9/4$ ) et si la donnée de Cauchy est nulle pour x assez grandalors la solution est nulle. Ce résultat est la conséquence d'uneinégalité de Carleman adaptée à la décroissance de la solution à l'infini.