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Documents  Le Rousseau, Jérôme | enregistrements trouvés : 16

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Research schools;Partial Differential Equations

This is a survey talk about the Boundary Control method. The method originates from the work by Belishev in 1987. He developed the method to solve the inverse boundary value problem for the acoustic wave equation with an isotropic sound speed. The method has proven to be very versatile and it has been applied to various inverse problems for hyperbolic partial differential equations. We review recent results based on the method and explain how a geometric version of method works in the case of the wave equation for the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. This is a survey talk about the Boundary Control method. The method originates from the work by Belishev in 1987. He developed the method to solve the inverse boundary value problem for the acoustic wave equation with an isotropic sound speed. The method has proven to be very versatile and it has been applied to various inverse problems for hyperbolic partial differential equations. We review recent results based on the method and explain how a ...

35R30 ; 35L05 ; 35L20

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Outreach;Mathematics Education and Popularization of Mathematics

Professeur à l'université de Versailles-Saint-Quentin
Président de l'association Animath

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Outreach;Mathematics Education and Popularization of Mathematics

Professeure émérite de Mathématiques Université de Rennes 1
Présidente du "Committe for Women in Mathematics"

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Outreach;Mathematics Education and Popularization of Mathematics

Professeur à l’Université Pierre et Marie Curie et au département de Mathématiques et applications de l’ENS
Membre du Laboratoire Jacques-Louis Lions
Laure Saint-Raymond a donné une conférence lors du premier Congrès de la Société Mathématique sur le thème "Echangeabilité, chaos et dissipation dans les grands systèmes de particules".

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Outreach;Mathematics Education and Popularization of Mathematics

Directeur de recherche au CNRS
Institut Fourier - Université de Grenoble 1

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Outreach;Mathematics Education and Popularization of Mathematics

Professeur des universités à l’université d’Orléans
Membre du Laboratoire MAPMO
CNRS, Université d'Orléans

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Outreach;Mathematics Education and Popularization of Mathematics

Mathématicien
Inspecteur Général de l'éducation nationale

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Outreach;Mathematics Education and Popularization of Mathematics

Directeur de l’Institut Henri Poincaré
Lauréat de la médaille Fields en 2010

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Research talks;Control Theory and Optimization;Partial Differential Equations

In this talk we present a inequality obtained with Jérôme Le Rousseau, for sum of eigenfunctions for bi-Laplace operator with clamped boundary condition. These boundary conditions do not allow to reduce the problem for a Laplacian with adapted boundary condition. The proof follow the strategy used for Laplacian, namely we consider a problem with an extra variable and we prove Carleman estimates for this new problem. The main difficulty is to obtain a Carleman estimate up to the boundary. In this talk we present a inequality obtained with Jérôme Le Rousseau, for sum of eigenfunctions for bi-Laplace operator with clamped boundary condition. These boundary conditions do not allow to reduce the problem for a Laplacian with adapted boundary condition. The proof follow the strategy used for Laplacian, namely we consider a problem with an extra variable and we prove Carleman estimates for this new problem. The main difficulty is to ...

35B45 ; 35S15 ; 93B05 ; 93B07

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Research schools;Partial Differential Equations


inverse problem - reconstruction - regularization - tomography - computation

65N21 ; 65N20 ; 35R25

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Research schools;Partial Differential Equations


inverse problem - reconstruction - regularization - tomography - computation

65N21 ; 65N20 ; 35R25

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Research talks;Control Theory and Optimization;Partial Differential Equations

We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get rapid stabilization for Korteweg-de Vries equations and how to stabilize in finite time $1-D$ parabolic linear equations by means of periodic time-varying feedback laws. We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get ...

35B35 ; 35Q53 ; 93C10 ; 93C20 ; 35K05 ; 93B05 ; 93B17 ; 93B52

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Research talks;Control Theory and Optimization;Partial Differential Equations

We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We show the strong convergence of the approximation and then discussed several examples for $N = 1$ and $N = 2$. The reconstruction of both the state and the source term is also discussed, as well as the boundary case. The parabolic case - more delicate as it requires the use of appropriate weights - will be also addressed. Joint works with Nicolae Cîndea and Diego Araujo de Souza. We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic ...

35L10 ; 65M12 ; 93B40

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Research talks;Control Theory and Optimization;Partial Differential Equations

We consider spectral optimization problems of the form

$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$

where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

$\lambda_1(\Omega;D)=\min\lbrace\int_\Omega|\nabla u|^2dx+k\int_{\partial D}u^2d\mathcal{H}^{d-1}:$

$u\in H^1(D),u=0$ on $\partial\Omega\cap D,||u||_{L^2(\Omega)}=1\rbrace$

reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains. The case of Dirichlet condition on a $\textit{fixed}$ part and of Neumann condition on the $\textit{free}$ part of the boundary is also considered
We consider spectral optimization problems of the form

$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$

where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation

$\lam...

49Q10 ; 49J20 ; 49N45

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Research talks;Control Theory and Optimization;Partial Differential Equations

We consider the problem of lagrangian controllability for two models of fluids. The lagrangian controllability consists in the possibility of prescribing the motion of a set of particle from one place to another in a given time. The two models under view are the Euler equation for incompressible inviscid fluids, and the quasistatic Stokes equation for incompressible viscous fluids. These results were obtained in collaboration with Thierry Horsin (Conservatoire National des Arts et Métiers, Paris) We consider the problem of lagrangian controllability for two models of fluids. The lagrangian controllability consists in the possibility of prescribing the motion of a set of particle from one place to another in a given time. The two models under view are the Euler equation for incompressible inviscid fluids, and the quasistatic Stokes equation for incompressible viscous fluids. These results were obtained in collaboration with Thierry Horsin ...

35Q93 ; 35Q31 ; 76D55 ; 93B05

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- vii-344 p.
ISBN 978-3-642-27892-1

Lecture notes in mathematics , 2048

Localisation : Collection 1er étage

controlabilité # stabilisation par sytème feedback # contrôle de flux # convergence # méthode numérique

93-06 ; 93B05 ; 93B07 ; 93C20 ; 93D15 ; 35B35 ; 65M12 ; 76B75 ; 00B25

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