Publications

2011

• Entanglement creation in low-energy scattering
  
  • Weder Ricardo
  Physical Review A : Atomic, molecular, and optical physics [1990-2015], American Physical Society, 2011, 84 (6). We study the entanglement creation in the low-energy scattering of two particles in three dimensions, for a general class of interaction potentials that are not required to be spherically symmetric. The incoming asymptotic state, before the collision, is a product of two normalized Gaussian states. After the scattering, the particles are entangled. We take as a measure of the entanglement the purity of one of them. We provide a rigorous explicit computation, with error bound, of the leading order of the purity at low energy. The entanglement depends strongly on the difference of the masses. It takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses. It is quite remarkable that the anisotropy of the potential gives no contribution to the leading order of the purity, in spite of the fact that entanglement is a second-order effect. © 2011 American Physical Society. (10.1103/physreva.84.062320)
  DOI : 10.1103/physreva.84.062320
• Second order analysis of optimal control problems with singular arcs. Optimality conditions and shooting algorithm.
  
  • Aronna Maria Soledad
  , 2011. This thesis deals with optimal control problems for systems that are affine in one part of the control variable. First, we state necessary and sufficient second order conditions when all control variables enter linearly. We have bound control constraints and a bang-singular solution. The sufficient condition is restricted to the scalar control case. We propose a shooting algorithm and provide a sufficient condition for its local quadratic convergence. This condition guarantees the stability of the optimal solution and the local quadratic convergence of the algorithm for the perturbed problem in some cases. We present numerical tests that validate our method. Afterwards, we investigate an optimal control problems with systems that are affine in one part of the control variable. We obtain second order necessary and sufficient conditions for optimality. We propose a shooting algorithm, and we show that the sufficient condition just mentioned is also sufficient for the local quadratic convergence. Finally, we study a model of optimal hydrothermal scheduling. We investigate, by means of necessary conditions due to Goh, the possible occurrence of a singular arc.
• A Stochastic Dynamic Principle for Hybrid Systems with Execution Delay and Decision Lags
  
  • Aouchiche K.
  • Bonnans J. Frederic
  • Granato Giovanni
  • Zidani Hasnaa
  , 2011, pp.6788-6793. This work presents a stochastic dynamic programming (SDP) algorithm that aims at minimizing an economic criteria based on the total energy consumption of a range extender electric vehicle (REEV). This algorithm integrates information from the REEV's navigation system in order to obtain some information about future expected vehicle speed. The model of the vehicle's energetic system, which consists of a high-voltage (HV) battery, the main energy source, and an internal combustion engine (ICE), working as an auxiliary energy source), is written as a hybrid dynamical system and the associated optimization problem in the hybrid optimal control framework. The hybrid optimal control problem includes two important physical constraints on the ICE, namely, an activation delay and a decision lag. Three methods for the inclusion of such physical constraints are studied. After introducing the SDP algorithm formulation we comment on numerical results of the stochastic algorithm and its deterministic counterpart.
• Mathematical analysis of the junction of two acoustic open waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Goursaud Benjamin
  • Hazard Christophe
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2011, 71 (6), pp.2048-2071. The present paper concerns the scattering of a time-harmonic acoustic wave by the junction of two open uniform waveguides, where the junction is limited to a bounded region. We consider a two-dimensional problem for which wave propagation is described by the scalar Helmholtz equation. The main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave. We use here modal radiation conditions which extend the classical conditions used for closed waveguides. They are based on the generalized Fourier transforms which diagonalize the transverse contributions of the Helmholtz operator on both sides of the junction. We prove the existence and uniqueness of the solution, which seems to be the first result in this context. The originality lies in the proof of uniqueness, which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized Fourier variable. © 2011 Society for Industrial and Applied Mathematics. (10.1137/100811374)
  DOI : 10.1137/100811374
• Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters
  
  • Desilles Anna
  , 2011. We present a few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems when the Hamiltonian (fundamental diagram) depends on time, position and/or some regulation parameters. We study such a problem in its equivalent variational formulation. In this case, the corresponding lagrangian depends on the state of the characteristic dynamical system. As the Lax-Hopf formulae that give the solution in a semi-explicit form for an homogeneous lagrangian do not hold, we use a capture basin algorithm to compute the Moskowitz function as a viability solution of the Hamilton-Jacobi-Moskowitz problem with general conditions (including initial, boundary and internal conditions). We present two examples of applications. In the first one we introduce the variable speed limit as a regulation parameter. Our approach allows to compute the Moscowitz function for all values of the variable speed limit in a selected range and then to analyze its influence on the traffic flow. In particular, we study the case when the variation of the speed limit is applied locally, in space and time. Our second example deals with the local load capacity variations on the road. Such a variation can be a permanent property of the road (road narrowing) or it can be due to a temporary change of number of lanes (an accident or roadworks, for example). One can also use it as a regulation parameter by variable assignment of a supplementary lane.
• Asymptotic expansions for interior solutions of semilinear elliptic problems
  
  • Bonnans J. Frederic
  • Silva Francisco J.
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2011, 49 (6), pp.2494-2517. In this work we consider the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated parametrized family of penalized problems, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to obtain some specific error bounds in various norms and for a general class of barrier functions. In this manner, we generalize the results of the previous work which were obtained in the ODE framework.
• Global optimization of pipe networks by the interval analysis approach: the Belgium network case
  
  • Bonnans Joseph Frederic
  • Spiers Grégoire
  • Vie Jean-Léopold
  , 2011, pp.13. We show that global optimization techniques, based on interval analysis and constraint propagation, succeed in solving the classical problem of optimization of the Belgium gas network.
• Optimal multiple stopping problem and financial applications
  
  • Ben Latifa Imene
  • Bonnans Joseph Frederic
  • Mnif Mohamed
  , 2011, pp.30. In their paper [2], Carmona and Touzi have studied an optimal multiple stopping time problem in a market where the price process is continuous. In this paper, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. Then we relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman Variational Inequality.
• Analyse mathématique et numérique de quelques problèmes d'ondes en milieu périodique
  
  • Coatléven Julien
  , 2011. De nombreux problèmes physiques sont modélisés par des équations aux dérivées partielles posées dans un domaine pour lesquels la géométrie ainsi que les coefficients sont décrits par des fonctions périodiques, hormis dans certaines régions de taille modeste par rapport à celle du domaine d'intérêt (on parle alors de perturbations pour ces régions). Les caractéristiques du problème sortant très souvent du cadre d'application des méthodes d'homogénéisation, nous avons développé des méthodes alternatives tirant parti de la periodicité afin de restreindre le domaine de calcul à des domaines bornés. Pour cela, nous avons généralisé les approches de type Lippmann-Schwinger, ce qui nous permet de traiter le cas de défauts bornés ou le cas de défauts non bornés structurés, la difficulté tenant au fait que l'on ne dispose pas dans le cas d'un milieu périodique quelconque d'une représentation analytique de la solution en l'absence de perturbation (i.e la fonction de Green est inconnue en général). Notre approche repose sur la connaissance des opérateurs de Dirichlet- to-Neumann (DtN) de bandes périodiques non bornés dans une seule direction. Nous traitons deux grandes familles de problèmes, les problèmes harmoniques, pour lesquels les opérateurs DtN dans les bandes sont connus, et les problèmes d'évolution, pour lesquels nous proposons une méthode de construction de ces opérateurs. Nous traitons dans ces deux situations le cas d'une perturbation bornée ou non, puis nous généralisons les techniques de scattering multiple du milieu homogène au cas périodique, afin de pouvoir traiter le cas de plusieurs perturbations.
• High order transmission conditions for thin conductive sheets in magneto-quasistatics
  
  • Schmidt Kersten
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2011, 45 (6), pp.1115-1140. We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses $\eps$ are essentially smaller or at the order of the skin depth. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t.~to the small parameter $\eps$ and obtain optimal bound for the modelling error outside the sheet of order $\eps^{N+1}$ for the condition of order N. Numerical experiments with high order finite elements for sheets with varying curvature verify the theoretical findings. (10.1051/m2an/2011009)
  DOI : 10.1051/m2an/2011009
• Mathematical and numerical modeling of wave propagation in fractal trees
  
  • Joly Patrick
  • Semin Adrien
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2011, 349 (19-20), pp.1047-1051. We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. The emphasis is put on the construction and approximation of transparent boundary conditions. © 2011. (10.1016/j.crma.2011.09.008)
  DOI : 10.1016/j.crma.2011.09.008
• Simulation tool for morphological analysis
  
  • Fronville Alexandra
  • Harrouet Fabrice
  • Desilles Anna
  • de Loor Pierre
  , 2010, pp.127--132. To understand the mechanisms underlying the morphogenesis of multicellular organisms we study the dynamic system of cells (cell multiplication, cell migration, apoptosis); local interactions between cells for understanding the convergence of the system to a stable form that is constantly renewed.and the controls established by the nature of the growth of the organism, and its convergence to a stable form. We must be able to formalize it in a proper metric space a metaphor of cell dynamics to nd conditions (decisions, states) in which operational constraints (such as those induced by the tissue or the use of resources) are always satis ed and therefore in which the system is viable and maintain its shape while renewing. The aim of this paper is to explain the mathematical foundations of this work and describe a simulation tool to study the morphogenesis of a virtual organism and to describe a simulation tool to study the morphogenesis of a virtual multicellular organism. We formalize mathematically a model of cell dynamic on the principles of morphological analysis. Morphological analysis and viability theory are the mathematical foundations that motivate this work and this tool will test whether a system generated by morphological equations can maintain its shape and remains "viable" in a given environment.
• Monotonicity condition for the $\theta$-scheme for diffusion equations
  
  • Bonnans J. Frederic
  • Tan Xiaolu
  , 2011, pp.6. We derive the necessary and sufficient condition for the $L^{\infty}-$monotonicity of finite difference $\theta$-scheme for a diffusion equation. We confirm that the discretization ratio $\Delta t = O(\Delta x^2)$ is necessary for the monotonicity except for the implicit scheme. In case of the heat equation, we get an explicit formula, which is weaker than the classical CFL condition.
• Gravitation classique
  
  • Perez Jérôme
  , 2011, pp.243 Pages. Cet ouvrage est un manuel de gravitation classique concrétisant de nombreuses années d'enseignement de l'auteur. Il se décompose en deux parties : les systèmes contenant très peu de corps massifs - typiquement deux, plus éventuellement des perturbations, et les systèmes en comportant beaucoup, suffisamment pour utiliser les techniques de la physique statistique. La dérivation des équations de la physique associées à ces différents problèmes permet de comprendre leurs relations intimes et leurs différentes origines tant historiques que techniques. L'application de ces divers formalismes au système solaire puis aux essaims stellaires que sont les amas globulaires ou les galaxies permet de mieux appréhender ces constituants importants de notre Univers.
• Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras
  
  • Pirio Luc
  • Russo Francesco
  , 2011. For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric Cremona transformations, proving also a structure theorem for these birational maps and for varieties 3-covered by twisted cubics by reinterpreting for these objects the solvability of the radical of a Jordan algebra.
• On projective varieties $n$-covered by curves of degree $\delta$
  
  • Pirio Luc
  • Russo Francesco
  , 2011. As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such that through $n\geq 2$ general points there passes an irreducible curve $C$ of degree $\delta\geq n-1$ we have $N\leq \pi(r,n,\delta+r(n-1)+2)$, where $\pi(r,n,d)$ is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety $Y^r\subset\mathbb P^{n+r-1}$ of degree $d$. A lot of examples of varieties as in the title and attaining the previous bound for the embedding dimension are constructed from Castelnuovo varieties and were thus dubbed {\it of Castelnuovo type} in [PT], where it is also proved that all extremal varieties as above are of this kind, except possibly when $n>2$, $r>1$ and $\delta=2n-3$. One of the main results of the paper is the classification of extremal varieties $X^{r+1}\subset \mathbb P^{2r+3}$ 3-covered by twisted cubics and not of Castelnuovo type. Interesting examples are provided by the so called {\it twisted cubics over complex Jordan algebras of rank 3}, as pointed out by Mukai. By relating to an extremal variety 3-covered by twisted cubics, via tangential projection, a quadro-quadric Cremona transformation in $\mathbb P^r$ we are able to classify all these object either for $r\leq 4$ or under the smoothness assumption. In the last case we obtain that they are either smooth rational normal scrolls (hence of Castelnuovo type) or the Segre embeddings of $\p^1\times Q^r$ or one of the four Lagrangian Grassmannians. We end by discussing some open problems pointing towards the equivalence of these apparently unrelated objects: extremal varieties 3-covered by twisted cubics, quadro-quadric Cremona transformations of $\mathbb P^r$ and complex Jordan algebras of dimension $r+1$ and of rank three.
• Numerical MicroLocal Analysis in Time domain
  
  • Collino Francis
  • Marmorat Simon
  , 2011. Ce rapport présente une étude de la méthode NMLA á partir de données dans le domaine temporel.
• Stabilité et commande des systèmes dynamiques
  
  • Jean Frédéric
  , 2011.
• Sur le calcul numérique des modes non linéaires
  
  • Blanc François
  • Ege Kerem
  • Touzé Cyril
  • Mercier Jean-François
  • Bonnet-Ben Dhia Anne-Sophie
  , 2011, pp.1-6. Nos travaux portent sur le calcul numérique de modes non linéaires. L’approche adoptée consiste à résoudre par différences finies l’équation aux dérivées partielles (EDP) décrivant la géométrie d’un mode non linéaire dans l’espace des phases. Cette EDP est vue comme un problème de transport dont on recherche les conditions initiales donnant des solutions périodiques. Les algorithmes de résolution et d’optimisation sont testés sur un système mécanique à deux degrés de liberté et à non linéarité cubique. Cet exemple nous permet de discuter de la convergence des algorithmes et des problèmes de mise en œuvre. Les résultats sont également comparés à des calculs par continuation.
• Optimal control problems of BV trajectories with pointwise state constraints
  
  • Forcadel Nicolas
  • Rao Zhiping
  • Zidani Hasnaa
  , 2011, 18. This paper deals with some optimal control problems governed by ordinary differential equations with Radon measures as data and with pointwise state constraints. We study the properties of the value function and obtain its characterization by means of an auxiliary control problem of absolutely continuous trajectories. For this, we use some known techniques of reparametrization and graph completion. We are also interested in the characterization of the value function as the unique constrained viscosity solution of a Hamilton-Jacobi equation with measurable time dependant Hamiltonians. (10.3182/20110828-6-IT-1002.01694)
  DOI : 10.3182/20110828-6-IT-1002.01694
• Diffusion as a singular homogenization of the Frenkel-Kontorova model
  
  • Alibaud Nathaël
  • Briani Ariela
  • Monneau Régis
  Journal of Differential Equations, Elsevier, 2011, 251 (4-5), pp.785-815. In this work, we consider a general fully overdamped Frenkel-Kontorova model. This model describes the dynamics of a infinite chain of particles, moving in a periodic landscape. Our aim is to describe the macroscopic behavior of this system. We study a singular limit corresponding to a high density of particles moving in a vanishing periodic landscape. We identify the limit equation which is a nonlinear diffusion equation. Our homogenization approach is done in the framework of viscosity solutions. (10.1016/j.jde.2011.05.020)
  DOI : 10.1016/j.jde.2011.05.020
• Indirect controllability of locally coupled wave-type systems and applications
  
  • Alabau-Boussouira Fatiha
  • Léautaud Matthieu
  , 2011. We consider symmetric systems of two wave-type equations only one of them being controlled. The two equations are coupled by zero order terms, localized in part of the domain. We prove an internal and a boundary null-controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. We deduce similar null-controllability results in any positive time for parabolic systems and Schrödinger-type systems under the same geometric conditions on the coupling and the control regions. This includes several examples in which these two regions have an empty intersection.
• Modeling and numerical simulation of a grand piano.
  
  • Chabassier Juliette
  • Joly Patrick
  , 2011, pp.00. We consider a complete model of a piano which accounts for the acoustical behavior of the instrument from excitation to soundand, and we propose a numerical discretisation. The model is described as well as the numerical methods used for its discretisation. Nonlinearities and couplings are treated in such a way that energy techniques ensure numerical stability. Numerical results are presented and compared to measurements.
• Identifying cracks in homogeneous and bimaterial bodies using 3D elastodynamic topological sensitivity
  
  • Bellis Cédric
  • Bonnet Marc
  , 2011.
• Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions
  
  • Graber Philip Jameson
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2011, 74, pp.3137-3148. We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions. (10.1016/j.na.2011.01.029)
  DOI : 10.1016/j.na.2011.01.029