Publications

2014

• La singularité voilée
  
  • Perez Jérôme
  • Alimi Jean-Michel
  Pour la science, Pour la Science, 2014, Dossier N°83, pp.p 123. La relativité générale prédit l'existence de points de densité infinie où les lois de la physique s'effondrent : les singularités. Certaines sont tapies au cœur des trous noirs et nous ne pouvons les observer. Qu'en est-il de la singularité initiale, le Big Bang ? Elle est, elle aussi, isolée, car la relativité générale masque cet étrange événement en introduisant le chaos à l'origine de l'Univers.
• Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations
  
  • Bayen Térence
  • Bonnans J. Frederic
  • Silva Francisco J.
  Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (4), pp.2063--2087. In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ growths quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions. (10.1090/S0002-9947-2013-05961-2)
  DOI : 10.1090/S0002-9947-2013-05961-2
• Optimal control problems on stratifiable state constraints sets.
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  , 2014. We consider an infinite horizon problem with state constraints K : inf Z 1 0 e t'(yx;u(t); u(t))dt u : [ 0 ;+1) ! A measurable yx;u(t) 2 K 8t 0 (P) : where > 0 is fixed and yx;u( ) is a trajectory of the control system ( y_ = f (y; u) a.e. t 0 y(0) = x 2 K We are mainly concerned with a characterization of the value function of (P) as the bilateral solution to a Hamilton-Jacobi-Bellman equation.
• Second-order sufficient conditions for strong solutions to optimal control problems
  
  • Bonnans Joseph Frederic
  • Dupuis Xavier
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (03), pp.704-724. In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem. (10.1051/cocv/2013080)
  DOI : 10.1051/cocv/2013080
• T-coercivity for the Maxwell problem with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  Communications in Partial Differential Equations, Taylor & Francis, 2014. In this paper, we study the time-harmonic Maxwell problem with sign-changing permittivity and/or permeability, set in a domain of R^3. We prove, using the T-coercivity approach, that the well-posedness of the two canonically associated scalar problems, with Dirichlet and Neumann boundary conditions, implies the well-posedness of the Maxwell problem. This allows us to give simple and sharp criteria, obtained in the study of the scalar cases, to ensure that the Maxwell transmission problem between a classical dielectric material and a negative metamaterial is well-posed.
• La clé du mystère de la lettre H ?
  
  • Perez Jérôme
  Images des mathématiques, CNRS, 2014. En physique théorique, en mécanique quantique, en optimisation, et dans bien d'autres domaines la lettre $H$ est traditionnellement rattachée à $H\!$amilton à travers le terme hamiltonien. Lorsque l'on fait l'exégèse de cette notation on constate pourtant que la notation lui est antérieure et a été introduite par Lagrange dans un contexte où $H\!$uygens semble être mis en avant... Un manuscrit redécouvert récemment dans l'un des ouvrages de la seconde édition de la mécanique analytique, publié par Lagrange en 1815 alors qu'Hamilton n'avait pas 10 ans, pourrait bien être la clé de ce mystère.
• Les fameux points de Lagrange -- Fameux, pour qui les connaît !
  
  • Perez Jérôme
  Images des mathématiques, CNRS, 2014. Parmi tous les domaines abordés par Joseph-Louis Lagrange la mécanique céleste tient une place de choix. C'est pendant sa période berlinoise, de 1766 à 1788 qu'il découvre une famille de points d'équilibre de certaines extensions du problème des deux corps. Les points de Lagrange étaient nés ! Depuis cette époque, nombreuses sont les extensions de cette théorie à différentes configurations. Et nombreuses sont aussi les observations astronomiques en relation directe avec ces théories.
• Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control
  
  • Fabbri Giorgio
  • Russo Francesco
  , 2014. The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$, taking values in a Hilbert space $H$, is the sum of a local martingale and a suitable "orthogonal" process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of "mild solution" for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.
• Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian
  
  • Graber P. Jameson
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2014, 70 (2), pp.185-224. We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed. Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE, viscosity solutions, front propagation, mean field games (10.1007/s00245-014-9239-3)
  DOI : 10.1007/s00245-014-9239-3
• Numerical modeling of nonlinear acoustic waves in a tube connected with an array of Helmholtz resonators
  
  • Lombard Bruno
  • Mercier Jean-François
  Journal of Computational Physics, Elsevier, 2014, 259 (15). (10.1016/j.jcp.2013.11.036)
  DOI : 10.1016/j.jcp.2013.11.036
• On the absence of trapped modes in locally perturbed open waveguides
  
  • Hazard Christophe
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2014, pp.14. This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the three-dimensional scalar wave equation is considered. (10.1093/imamat/hxu046)
  DOI : 10.1093/imamat/hxu046
• Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time Effects
  
  • Abdulle Assyr
  • Grote Marcus J.
  • Stohrer Christian
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2014, 12 (3), pp.1230–1257. A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our new FE-HMM-L method captures not only the short-time behavior of the wave field, well described by classical homogenization theory, but also more subtle long-time dispersive effects, both at a computational cost independent of the microscale. Optimal error estimates in the energy norm and the $L^2$-norm are proved over finite time intervals, which imply convergence to the solution from classical homogenization theory when both the macro- and the microscale are refined simultaneously. Numerical experiments illustrate the usefulness of the FE-HMM-L method and corroborate the theory. (10.1137/13094195X)
  DOI : 10.1137/13094195X
• Extraordinary transmission through subwavelength dielectric gratings in the microwave range
  
  • Ahmed Akarid
  • Ourir Abdelwaheb
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  Optics Letters, Optical Society of America - OSA Publishing, 2014, 39 (13), pp.3752-3755. We address the problem of the transmission through subwavelength dielectric gratings. Following Maurel et al. [Phys. Rev. B 88, 115416 (2013)], the problem is reduced to the transmission by an homogeneous slab, either anisotropic (for transverse magnetic waves, TM) or isotropic (for transverse electric waves, TE), and an explicit expression of the transmission coefficient is derived. The optimum angle realizing perfect impedance matching (Brewster angle) is shown to depend on the contrasts of the dielectric layers with respect to the air. Besides, we show that the Fabry–Perot resonances may be dependent on the incident angle, in addition to the dependence on the frequency. These facts depart from the case of metallic gratings usually considered; they are confirmed experimentally both for TE and TM waves in the microwave regime. (10.1364/OL.39.003752)
  DOI : 10.1364/OL.39.003752
• Quadro-quadric cremona transformations in low dimensions via the JC-correspondence
  
  • Pirio Luc
  • Russo Francesco
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2014, 64 (1), pp.71-111. We apply the results of arXiv:1109.3573 to study quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension four and five. (10.5802/aif.2839)
  DOI : 10.5802/aif.2839
• Asymptotic behaviour of codes in rank metric over finite fields
  
  • Loidreau P
  Designs, Codes and Cryptography, Springer Verlag, 2014, 71 (1), pp.105-118. In this paper, we rst recall some basic facts about rank metric. We then derive an asymptotic equivalent of the minimum rank distance of codes that reach the rank metric GilbertVarshamov bound. We then derive an asymptotic equivalent of the average minimum rank distance of random codes. We show that random codes reach GV bound. Finally, we show that optimal codes in rank metric have a packing density which is bounded by functions depending only on the base eld and the minimum distance and show the potential interest in cryptographic applications. (10.1007/s10623-012-9716-0)
  DOI : 10.1007/s10623-012-9716-0
• Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds
  
  • Ghezzi Roberta
  • Jean Frédéric
  , 2014, 5, pp.201-218. (10.1007/978-3-319-02132-4_13)
  DOI : 10.1007/978-3-319-02132-4_13
• Improved multimodal method in varying cross section waveguides
  
  • Maurel Agnes
  • Mercier Jean-François
  • Pagneux Vincent
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 470, pp.20130448. An improved version of the multimodal admittance method in acoustic waveguides with varying cross sections is presented. This method aims at a better convergence with respect to the number of transverse modes that are taken into account. It is based on an enriched modal expansion of the pressure: the N first modes are the local transverse modes and a supplementary (N+1)th mode, called boundary mode, is a well-chosen transverse function orthogonal to the N first modes. This expansion leads to the classical form of the coupled mode equations where the component of the boundary mode is of evanescent character. Under this form, the multimodal admittance method based on the Riccati equation on the admittance matrix (the Dirichlet-to-Neumann operator) is straightforwardly implemented. With this supplementary mode, in addition to the improvement of the convergence of the pressure field, results show a superconvergence of the scattered field outside of the varying cross sections region. (10.1098/rspa.2013.0448)
  DOI : 10.1098/rspa.2013.0448
• Quick reachability and proper extension for problems with unbounded controls
  
  • Aronna Maria Soledad
  • Motta Monica
  • Rampazzo Franco
  , 2014. For a CONTROL SYSTEM of the form _ x = f (x; u; v) + Σm =1 g (x)u_ ; on [0;T]; (x; u)(0) = ( x; u); with x : [0;T] ! IRn; u : [0;T] ! U IRm; v : [0;T] ! V IRl ; we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc.
• Finite element computation of trapped and leaky elastic waves in open stratified waveguides
  
  • Treyssede Fabien
  • Nguyen Khac-Long
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  Wave Motion, Elsevier, 2014, 51 (7), pp.pp.1093-1107. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results. (10.1016/j.wavemoti.2014.05.003)
  DOI : 10.1016/j.wavemoti.2014.05.003
• An approximation scheme for an Eikonal Equation with discontinuous coefficient
  
  • Festa Adriano
  • Falcone Maurizio
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2014, 52 (1), pp.236-257.. We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a-priori error estimate for the scheme in an integral norm. The last section contains some applications to control and image processing problems. (10.1137/120901829)
  DOI : 10.1137/120901829
• Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations
  
  • Rao Zhiping
  • Siconolfi Antonio
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2014, 257 (11), pp.3978--4014. We establish a comparison principle for a Hamilton-Jacobi-Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of "-partition and minimal "-partition for intervals of definition of an integral trajectory. (10.1016/j.jde.2014.07.015)
  DOI : 10.1016/j.jde.2014.07.015
• Generalized covariation for Banach space valued processes, Itô formula and applications
  
  • Di Girolami Cristina
  • Russo Francesco
  Osaka Journal of Mathematics, Osaka University, 2014, 51 (3). This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
• Propagation in waveguides with varying cross-section and curvature: A new light on the role of supplementary modes in multimodal methods
  
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 470 (2166), pp.20130743. We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems. (10.1098/rspa.2014.0008)
  DOI : 10.1098/rspa.2014.0008
• Level-set approach for Reachability Analysis of Hybrid Systems under Lag Constraints
  
  • Granato Giovanni
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (1), pp.606--628. This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite differences schemes for solving the system of inequalities is proposed and the convergence of the numerical solution towards the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application, namely, that of range extender electric vehicles. This class of electric vehicles uses an additional module -- the range extender -- as an extra source of energy in addition to its main source -- a high voltage battery. The reachability study of this system is used to establish the maximum range of a simple vehicle model. (10.1137/120874205)
  DOI : 10.1137/120874205
• BSDEs under partial information and financial applications.
  
  • Ceci Claudia
  • Cretarola Alessandra
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2014. In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Föllmer-Schweizer decomposition of a square integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model. (10.1016/j.spa.2014.03.003)
  DOI : 10.1016/j.spa.2014.03.003