\n"; echo ""; echo $titre."\n

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  Lundi 1er Mardi 2 Mercredi 3 Jeudi 4 Vendredi 5
09h00-09h50 Accueil et séance inaugurale Reception of participants and Opening session
08h30-11h00		 #yoccoz">Jean-Christophe Yoccoz #caffarelli">Luis Caffarelli #souganidis">Panagiotis Souganidis #majda">Andrew Majda
10h00-10h50 #cani">Marie-Paule Cani #ghil">Michael Ghil #villani">Cédric Villani #brezzi">Franco Brezzi
11h30-12h20 #ball">John Ball #hou">Thomas Hou #rannacher">Rolf Rannacher #zuazua">Enrique Zuazua #nirenberg">Louis Nirenberg
14h30-15h20 #ambrosio">Luigi Ambrosio #coron">Jean-Michel Coron #baccelli">François Baccelli #perthame">Benoît Perthame  
15h30-16h20 #fink">Mathias Fink #faugeras">Olivier Faugeras #tadmor">Eitan Tadmor #patera">Anthony Patera
17h00-17h50 #vishik">Mark Vishik #papanicolaou">George Papanicolaou #chorin">Alexandre Chorin #evans">Lawrence Evans
18h00-18h50     #varadhan">Srinivasa Varadhan  


Accueil et séance inaugurale Reception of participants and Opening session

Programme jour par jour Day by day programme


The joint evolution of the throughputs obtained by a set TCP flows sharing a single link can be represented by iterates of random affine maps.
A general network contains an arbitrary number of links and various types of flows (as many as there are routes through the network links). In this case, the associated dynamical system can be seen either as iterates of piecewise affine maps, or as a stochastic billiards; the state space of this billiards is a polyhedron of the Euclidean space, with dimension the number of flows and with as many faces as there are links in the network.
The study of the stationary regimes of these dynamical systems allows one to analyze the bandwidth sharing operated by TCP and to quantify the fairness of the protocol. It also allows one to compute the autocorrelation function of the point process of losses, as well as the law of the fluctuations of the throughput obtained by each flow. This also leads to aggregated traffic models with local regularity statistical properties similar to those observed on real traffic.
Joint work with Dohy Hong (INRIA & ENS).

Un Modèle Probabiliste du Contrôle de Congestion dans l'Internet

Le protocole TCP (Transmission Control Protocol) est le mécanisme de contrôle distribué qui permet d'éviter la congestion dans l'Internet. Ce protocole réalise un partage adaptatif des ressources (bande passante des liens) entre tous les flots présents dans le réseau.
Dans le cas du partage d'un seul lien (modèle simplifié où on ne considère que le lien d'accès), l'évolution jointe des débits obtenus par un ensemble de flots TCP peut être représentée par l'itération d'applications affines en milieu aléatoire.
Un réseau général comprend un nombre quelconque de liens ainsi que divers types de flots (autant que de routes à travers les liens du réseau). Dans ce cas, le système dynamique associé peut être vu soit comme l'itération d'applications aléatoires affines par morceaux, soit comme un billard stochastique; l'espace d'état de ce billard est un polyèdre de l'espace euclidien, qui a pour dimension le nombre des flots et qui possède autant de faces qu'il y a de liens dans le réseau.
L'étude du régime stationnaire de ces systèmes dynamiques permet d'analyser le partage de la bande passante réalisé par TCP et notamment de quantifier l'équité du protocole. Elle permet de calculer l'autocorrélation du processus ponctuel des pertes ainsi que les fluctuations du débit pour chaque flot. On en déduit aussi des modèles de trafic agrégé dont les propriétés statistiques de régularité locale sont similaires à celles observées sur le trafic réel.
Travail en collaboration avec Dohy Hong."; affiche_resume(); $abr="ball"; $nomstatus="John Ball (Mathematical Institute, Oxford)"; $titre="The Euler-Lagrange equation and minimizers in elastostatics"; $resume="One-dimensional examples from the calculus of variations show that it is not always the case that minimizers are weak solutions of the corresponding Euler-Lagrange equation. Whether or not this is the case for nonlinear elasticity is an open problem. The lecture will revisit old work of the speaker, giving somewhat improved results concerning conditions under which certain weak forms of the equilibrium equations can be established. Some other outstanding open questions in the area will be reviewed."; affiche_resume(); $abr="brezzi"; $nomstatus="Franco Brezzi (Istituto di Analisi Numerica del CNR, Pavia)"; $titre=" Mathematical aspects of the Chimera methods"; $resume="The Chimera Method is a numerical technique based on the decomposition of the computational domain into (overlapping) subdomains, using then different (and totally unrelated) grids in each subdomain. The lecture will present an overview of the mathematical results that are available for this kind of methods."; affiche_resume(); $abr="caffarelli"; $nomstatus="Luis Caffarelli (University of Texas, Austin)"; $titre="Nonlinear equations in random media"; $resume="We will discuss several aspects of homogeneisation of fully nonlinear equations in a random media: What does it mean to find a limit, and how tools from the theory of elliptic problems with constraints can be used for that purpose."; affiche_resume(); $abr="cani"; $nomstatus="Marie-Paule Cani (INPG, Grenoble)"; $titre="Efficient Animation of Natural Scenes for Computer Graphics"; $resume="Modeling and animating natural objects and phenomena with enhanced visual realism is a fascinating challenge for Computer Animation. Moreover, doing this efficiently becomes essential for a variety of applications, from video games to special effects and feature films.
Although physically-based modeling can be used to simulate motion and deformation with guarentees of realism, directly applying them to these complex, heterogeneous scenes would requires tremendous amounts of computational time. We thus propose alternative models, based on the following methodology:
- we firstly part the phenomenon or the object to model into a hierarchy of interacting sub-models, possibly simulated at totally different space and time scales;
- any kind on model (physically-based model, geometry, texture) may be used to define each element of this hierarchy. We exploit the levels of detail paradigm to switch between these representations when necessary during the animation;
- lastly, we define adaptive physically-based models that automatically tune space and time sampling in order to concentrate the computations when and where needed the most.
We illustrate this methodology by presenting a variety of applications, from the modeling of parts of the human body (deformable organs, hair) to the animation of natural scenes (lava flows, prairies blowing in the wind).

Animation efficace de scènes naturelles pour la synthèse d'images

L'animation d'objets et de phénomènes naturels est un objectif passionnant pour l'informatique graphique. Le faire en combinant efficacité et réalisme visuel est rendu essentiel par la montée en puissance des applications des images de synthèse au domaine de l'audiovisuel (effets spéciaux, jeux vidéo, films d'animation).
Si les modèles physiques permettent, en théorie, de calculer mouvements et déformations de manière réaliste, simuler et visualiser des scènes complexes et hétérogènes en des temps raisonnables demeure extrêmement difficile. Il s'agit donc de mettre au point des modèles alternatifs, permettant de concentrer la puissance de calcul là où elle est la plus nécéssaire.
Nous discutons trois voies complémentaires pour y arriver :
- décomposer l'objet ou le phénomène en une hiérarchie de sous-modèles inter-agissants;
- avoir recours, pour chaque modèle de la hiérarchie, à des représentations adaptées mais minimalistes, du physique au géométrique en passant par de simples textures animées. Des transitions entre ces représentations sont éventuellement déclanchées au cours du calcul, selon le niveau de détail nécessaire;
- définir, pour les modèles les plus coûteux, des algorithmes d'animation adaptatifs ou multi-résolution, permettant de jouer sur le niveau de détail au cours de la simulation.
Nous illustrons ces techniques en présentant toute une variété d'applications, de l'animation de formes organiques (organes du corps humain, chevelures) à celle de scènes naturelles (coulées de lave, végétation animée)."; affiche_resume(); $abr="chorin"; $nomstatus="Alexandre Chorin (University of California, Berkeley)"; $titre="Conditional expectations and the renormalization group"; $resume="One often needs to estimate a partial set of variables in a problem too complex to be fully resolved, even when there is no significant gap in scales and frequencies between these variables and those that must be omitted. In \"optimal prediction\" methods one attempts to do this by calculating conditional expectations given an initial joint distribution for all the variables, and in renormalization group methods one uses a joint probability distribution to integrate out unwanted scales. I will exhibit a transformation that connects the two methods. This transformation produces the full parameter flow of a renormalization groupfrom small-cell Monte-Carlo calculations, and also permits the use of renormalization methods in prediction. Examples will be given."; affiche_resume(); $abr="coron"; $nomstatus="Jean-Michel Coron (Université Paris-Sud, Orsay)"; $titre="Return method and flow control"; $resume="During the nineties, J.-L. Lions has drawn the attention of mathematicians to flow control. In particular he has obtained various important results and has given several conjectures on the controllability of incompressible fluids. In this talk we present some results on these conjectures which have been obtained by means of the return method.
The return method, that we have introduced for a stabilization problem in finite dimension and first used in infinite dimension for the controllability of the Euler equations, allows in some cases to get the local controllability at an equilibrium of a nonlinear control system even if the linearized control system at the equilibrium is not controllable. The idea of the return method is the following one. If one can find a trajectory of the nonlinear control system such that
- it starts and ends at (or ``close enough to'') the equilibrium,
- the linearized control system around this trajectory is controllable,
then, in general, the inverse function theorem allows to conclude that one can go from any state close to the equilibrium to any other state close to the equilibrium. Hence the return method allows, in some cases, to reduce the problem of local controllability of a nonlinear control system to the controllability of linear systems. Let us recall that, for linear control systems, there are various methods to prove the controllability even in infinite dimension, in particular the powerful Hilbert Uniqueness Method (HUM) due to J.-L. Lions.
In this talk, we sketch some results in flow control which have been obtained by the return method, namely
- Global controllability results for the Euler equations of incompressible fluids,
- Global controllability results for the Navier-Stokes equations of incompressible fluids,
- Local controllability of a 1-D tank containing a fluid modeled by the shallow-water equation."; affiche_resume(); $abr="evans"; $nomstatus="Lawrence Craig Evans (University of California, Berkeley)"; $titre="PDE methods for weak KAM theory"; $resume="I will discuss some recent variational and PDE methods that D. Gomes and I have developed to study the \"weak KAM\" theory of Mather and Fathi. I will explain the relevance of duality theory from linear programming and describe also a new approximate variational principle."; affiche_resume(); $abr="faugeras"; $nomstatus="Olivier Faugeras (INRIA, Sophia-Antipolis)"; $titre="On the well-posedness of several problems in computer vision"; $resume="Computer and biological vision are fantastic sources of very interesting problems in the area of the calculus of variations. Three-dimensional vision includes the capacity of recovering the third dimension and the motion of objects in a scene when multiple images of that scene are available. This capacity implies the solution of the problem of finding correspondences between these images. This iconic matching problem can easily be cast in a variational framework resulting in ordinary differential equations (ODEs) defined on some standard functional spaces such as Sobolev spaces. Except in some simplified cases these ODEs are not partial differential equations (PDEs) because of the way similarity is measured between images. We nonetheless show that binocular stereo, optical flow computation and the general iconic matching problem are usually well-posed. Examples are given on real images."; affiche_resume(); $abr="fink"; $nomstatus="Mathias Fink (ESPCI, Paris)"; $titre="Time-reversed Acoustics"; $resume="Time-reversal invariance is a very powerful concept in classical and quantum mechanics. In the field of Acoustics, where time reversal invariance also occurs, time-reversal experiments may be achieved simply with time reversal mirrors made of arrays of transmit-receive transducers, allowing an incident acoustic field to be sampled, recorded, time-reversed and re-emitted.
Time reversal mirrors are innovative tools in the field of fundamentals physics. They may be used to study random media, multiple scattering processes, chaotic scattering, inverse scattering problems, dissipation effects and diffraction limits. They open the way to new signal processings. An overview of these fields will be presented and various applications of these methods to imaging, telecommunications and medical therapy will be described."; affiche_resume(); $abr="ghil"; $nomstatus="Michael Ghil (UCLA, Los Angeles)"; $titre="Bifurcations and pattern formation in the atmosphere and oceans"; $resume="The global climate system is composed of a number of subsystems-atmosphere, biosphere, cryosphere, hydrosphere and lithosphere-each of which has distinct characteristic times, from days and weeks to centuries and millennia. Each subsystem has its own internal variability, all other things being constant. The nonlinearity of the feedbacks within each subsystem can result in the coexistence of stable equilibria, the presence of self-sustained oscillations, and the possibility of deterministic chaos. We discuss the bifurcation trees associated with large-scale atmospheric and oceanic flows, across a hierarchy of models, from the simplest and most idealized ones to fairly realistic ones. It is shown how the tools of bifurcation theory and of the ergodic theory of dynamical systems are applied to fully three-dimensional general circulation models, with considerable spatial detail and physical realism, and even to coupled ocean-atmosphere models."; affiche_resume(); $abr="hou"; $nomstatus="Thomas Hou (Caltech, Pasadena)"; $titre="Multiscale Modeling and Computation of Incompressible Flows"; $resume="Many problems of fundamental and practical importance contain multiple scale solutions. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. Here, we introduce a dynamic multiscale method for computing nonlinear partial differential equations with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions local in time, and use them to approximate the multiscale solution for large times. Such approach overcomes the common difficulty associated with the memory effect and the non-unqiueness in deriving the global averaged equations for incompressible flows with multiscale solutions. It provides an effective multiscale numerical method for computing incompressible Euler and Navier-Stokes equations with multiscale solutions. In a related effort, we introduce a new class of numerical methods to solve the stochastically forced Navier-Stokes equations. We will demonstrate that our numerical method can be used to compute accurately high order statitstical quantites more efficiently than the traditional Monte-Carlo method."; affiche_resume(); $abr="majda"; $nomstatus="Andrew Majda (New York University)"; $titre="From the Ocean to Jupiter to the Truncated Burgers Hopf Equation: Novel Applications and Mathematical issues for statistical mechanics"; $resume="Geophysical flows are a rich area for potential applications of statistical mechanics ideas with several competing theories attempting to assess in various ways (Kraichnan, 1975; Salmon, et al 1977; Onsager 1949; Miller 1990; Robert 1991, etc), information from the infinite list of invariant conserved quantities for the inviscid dynamics. Here a recently emerging modified new viewpoint on such theories and their potential application will be presented (Majda and Holen, 1997; Turkington, 1998; DiBattista, Majda, Grote 2001) together with recent applications to parametrizing deep ocean convection (DiBattista and Majda, 2000, 2002) and prediction of the jets and spots on Jupiter (Turkington, Majda, Haven, DiBattista, 2001). In particular, the new statistical theories self-consistently predict the detailed observational record of Jupiter based on the Voyager and Galileo space missions. In the second part of the talk, the statistical mechanics of suitable truncations of the Burgers-Hopf equations will be utilized as a simple model for statistical features of the atmosphere (Majda and Timofeyev, 2000, 2001) as well as a model for the issues of statistically relevant and irrelevant conserved quantities (Abramov, Kovacic, Majda, 2002)."; affiche_resume(); $abr="nirenberg"; $nomstatus="Louis Nirenberg (New York University)"; $titre="On the distance function to the boundary, cut locus and some Hamilton-Jacobi equations"; $resume="In joint work with Y.Y. Li, we consider a smooth domain on a complete Riemannian manifold and study thr distance function to the boundary. It is shown that the map from the boundary along the normal is Lipschitz continuous.This was recently proved, earlier, by J.-I. Itoh and M. Tanaka. Our proof is different and will be presented."; affiche_resume(); $abr="papanicolaou"; $nomstatus="George Papanicolaou (Stanford University)"; $titre="Time-reversal, imaging and communications in random media"; $resume="Array imaging in random media can provide information about the location of strong scatterers as well as the inhomogeneities. I will discuss how this remote sensing can be used in the performance of time reversal refocusing as well as in designing very reliable and spatially localized communication systems in randomly inhomogeneous environments."; affiche_resume(); $abr="patera"; $nomstatus="Anthony Patera (MIT, Cambridge MA)"; $titre="Reduced-Basis Output Bounds: Reliable Real-Time Solution of Parametrized Partial Differential Equations"; $resume="We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures - methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
In this talk we describe the computational formulation and associated numerical analysis; and we present several examples drawn from applied mechanics.
Work in collaboration with: S. Ali, L. Machiels, Y. Maday, C. Prud'homme, D. Rovas, G. Turinici, K. Veroy"; affiche_resume(); $abr="perthame"; $nomstatus="Benoît Perthame (Ecole Normale Supérieure, Paris)"; $titre="Mathematical questions along the flow of a river"; $resume="Motivated by computational aspects of Saint-Venant's shallow water sytem (usual for many applications like rivers flow, tidal waves, but also narrow tubes), we consider some mathematical and algorithmic questions for hyperbolic systems with a topography driven source term. We also revisit some classical questions in the numerical analysis of finite volume methods such as: what are sharp CFL conditions for E-schemes, why TVD bounds on the approximate solutions ARE NOT necessary for h1/2 convergence rates."; affiche_resume(); $abr="rannacher"; $nomstatus="Rolf Rannacher (Universität Heidelberg)"; $titre="Duality techniques in error control and optimization for PDEs"; $resume=" We present a systematic approach to error control and mesh adaptivity in the numerical solution of optimal control problems governed by partial differential equations. By the Lagrangian formalism the optimization problem is reformulated as a saddle-point boundary value problem that is discretized by a Galerkin finite element method. The accuracy of the discretization is controlled by residual-based a posteriori error estimates. The main features of this method will be illustrated by examples from optimal control of fluids and parameter estimation."; affiche_resume(); $abr="souganidis"; $nomstatus="Panagiotis Souganidis (University of Texas, Austin)"; $titre="Fully nonlinear stochastic partial differential equations: Theory and Applications"; $resume="I will present a new theory for fully nonlinear stochastic pde, which P.-L. Lions and I have have been developing over the last few years, and I will discuss its applications to stochastic control and phase transitions."; affiche_resume(); $abr="tadmor"; $nomstatus="Eitan Tadmor (UCLA, Los Angeles & University of Maryland, College Park)"; $titre="Critical Thresholds in Restricted Euler Dynamics"; $resume="We study the questions of global regularity vs. finite time breakdown in Eulerian dynamics, Dt U =grad F, which shows up in different contexts dictated by different modeling of F's. To address these questions, we propose the notion of Critical Threshold (CT), where a conditional finite time breakdown depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold. We shall outline three prototype cases. We begin with the Euler-Poisson equations, in one-dimension and in the case of multidimensional geometric symmetry, with or without forcing mechanisms of relaxation, viscosity,... Next, we extend our discussion to a range of genuinely multidimensional problems by tracing the eigenvalues of the gradient matrix, g := g(Ui,j), which are shown to be governed by Riccati equation, Dt g + g2 = (l,D2F r), involving the forcing eigenpair (l,r). Equipped with this description of the spectral dynamics we turn to the n-dimensional Restricted Euler equations. Here we obtain [n/2]+1 global invariants which enable us to precisely characterize the local topology at breakdown time, (extending previous studies in the n=3-dimensional case initiated by Vieillefosse). And finally, we introduce the corresponding n-dimensional Restricted Euler-Poisson (REP) system, identifying a set of [n/2] global invariants, which yield (i) sufficient conditions for finite time breakdown, and (ii) a remarkable characterization of two-dimensional initial REP configurations with global smooth solutions. Consequently, the CT in this case is shown to depend on the initial density, rho0, the initial divergence, div U0, and the initial spectral gap, g1(0) - g2(0). In this lecture we survey a series of our recent works with Hailiang Liu, UCLA."; affiche_resume(); $abr="varadhan"; $nomstatus="Srinivasa Varadhan (New York University)"; $titre="Large Deviations, variational formulas and nonlinear equations"; $resume="The theory of large deviations tries to estimate the probabilities of rare events. Often some form of entropy plays a role in the precise estimation in a logarithmic scale. This often leads to variational formulas and related non-linear equations. We will look at examples of this."; affiche_resume(); $abr="villani"; $nomstatus="Cédric Villani (Ecole Normale Supérieure, Lyon)"; $titre="H Theorem and convergence to equilibrium for solutions of the Boltzmann equation"; $resume="Boltzmann's H Theorem predicts the convergence of a rarefied gas, enclosed in a box, towards a state of thermodynamical equilibrium (maximum of entropy). When one is looking at a nice solution of the Boltzmann equation, satisfying certain uniform a priori bounds, then this guess is easy to justify mathematically by an elementary compactness argument. Considerably more tricky is the problem of finding explicit estimates about how fast convergence towards equilibrium occurs. This question, first raised by Kac in the fifties, can be connected with many fields of research: information theory, logarithmic Sobolev inequalities, and the study of compressible hydrodynamic equations. Here I shall present recent work on the subject, mostly done in collaboration with L. Desvillettes."; affiche_resume(); $abr="vishik"; $nomstatus="Mark Vishik ("; if ($lang=="fr") { $nomstatus=$nomstatus."Institut des problèmes de transmission de l'information, Moscou"; } else { $nomstatus=$nomstatus."Institute for Information Transmission Problems, Moscow"; } $nomstatus=$nomstatus.")"; $titre="Trajectory and Global Attractors for Evolutionary Equations"; $resume="We consider an abstract autonomous evolution equation. We do not suppose that the solution of the corresponding Cauchy problem is unique. Under natural conditions the trajectory attractor A for this equation is constructed. We consider a family K+ of its solutions {u(s), s>= 0}. The time translation semigroup {T(t), t>= 0} acts on K+: {T(t)u(s)=u(t+s)}. We assume that T(t)K+ is included in K+ (t>= 0) and T(t) is continous in the corresponding topology. The global attractor A of the semigroup {T(t)|K+} is called the trajectory attractor. We formulate the existence theorem for trajectory attractor of the evolutionary equation and describe its properties. In particular A is compact in C(R,E0), where E0 is a Banach space. We construct the global attractor A of the studied evolutionary equation using the trajectory attractor A. A coincides with the section of A at a fixed moment t, for example t=0 :
A=A(0)={u(0)| {u(t), t>= 0} belongs to A }.
We prove that A has the familiar properties of the global attractor.
Using the general scheme we construct the trajectory and the global atttractors for dissipative equations and systems of mathematical physics for which the uniqueness theorem for the Cauchy problem does not hold or is not proved yet. In the lecture we construct these attractors for the 3D Navier-Stokes system in a bounded domain. The most important part of the construction is a proper choice of a space K+ of solutions on R+, the choice of an appropriate topology of attraction to the attractor, and the construction of a compact absorbing set for the translation semigroup {T(t), t>= 0}|K+.
Trajectory and global attractors were constructed for some nonautonomous dissipative evolutionary equations. In the lecture we consider the nonautonomous 2D Navier-Stokes system with time dependend forcing term. Under some conditions the structure of this global attractor A will be given. For forcing term rapidly oscillating in time, we establish the quantative homogenization of the global attractor. All the results are obtained in collaboration with V. V. Chepyzhov."; affiche_resume(); $abr="yoccoz"; $nomstatus="Jean-Christophe Yoccoz (Collège de France, Paris)"; $titre="Homoclinic bifurcations of surface diffeomorphisms"; $resume="A homoclinic orbit in a dynamical system is an orbit whose past and future asymptotics is the same periodic orbit, but which is not itself periodic. Poincaré was the first to guess their crucial importance for understanding the complexity of the dynamics. We will survey the results obtained in the subject in the last twenty years, as well on the geometrical side (fractal dimensions) as the dynamical one (non-uniformly hyperbolic dynamics with critical tangencies).

Bifurcations homoclines des difféomorphismes des surfaces

Une orbite homocline dans un système dynamique est une orbite qui converge, dans le passé et le futur, vers une même orbite périodique, sans être elle-même périodique. H. Poincaré est le premier à avoir deviné le rôle crucial qu'elles jouent dans la compréhension de la complexité d'un système. Nous survolerons les progrès effectués ces vingt dernières années dans le sujet, aussi bien dans leur composante géométrique (importance cruciale des dimensions fractales) que dynamique (dynamique non-uniformément hyperbolique en présence de tangences critiques)."; affiche_resume(); $abr="zuazua"; $nomstatus="Enrique Zuazua (Universidad Complutense, Madrid)"; $titre="Controllability of some Partial Differential Equations"; $resume="There have been important progresses in the understanding of the controllability properties of Partial Differential Equations in the last fifteen years. This progress was motivated to a great extent by the HUM (Hilbert Uniqueness Method) introduced by J.-L. Lions. In this lecture we shall describe some of the most relevant results obtained on this problem. We shall also present some more recent work on hybrid systems coupling wave and heat equations arising in fluid-structure interaction and also on how the controllability property behaves under numerical discretizations for wave equations."; affiche_resume(); if (!($type=="resume")){ echo "
\n" ;} if ($lang=="fr"){ echo "Mise à jour le"; } else {echo "Last update";}?> 24 juin 2002 (fm ap)