Publications

2011

• Deterministic state constrained optimal control problems without controllability assumptions
  
  • Bokanowski Olivier
  • Forcadel Nicolas
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2011, 17 (4), pp.pp. 995-1015. In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position $x$. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems. (10.1051/cocv/2010030)
  DOI : 10.1051/cocv/2010030
• Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier Singular Complement Method
  
  • Ciarlet Patrick
  • Labrunie Simon
  Journal of Difference Equations and Applications, Taylor & Francis, 2011, 3, pp.113-155. We present an efficient method for computing numerically the solution to the time-dependent Maxwell equations in an axisymmetric domain, with arbitrary (not necessarily axisymmetric) data. The method is an extension of those introduced in [20] for Poisson's equation, and in [4] for Maxwell's equations in the fully axisymmetric setting (i.e., when the data is also axisymmetric). It is based on a Fourier expansion in the azimuthal direction, and on an improved variant of the Singular Complement Method in the meridian section. When solving Maxwell's equations, this method relies on continuous approximations of the fields, and it is both H(curl)- and H(div)-conforming. Also, it can take into account the lack of regularity of the solution when the domain features non-convex edges or vertices. Moreover, it can handle noisy or approximate data which fail to satisfy the continuity equation, by using either an elliptic correction method or a mixed formulation. We give complete convergence analyses for both mixed and non-mixed formulations. Neither refinements near the reentrant edges or vertices of the domain, nor cutoff functions are required to achieve the desired convergence order in terms of the mesh size, the time step and the number of Fourier modes used.
• Non-reflecting boundary conditions for acoustic propagation in ducts with acoustic treatment and mean flow
  
  • Redon Emmanuel
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  • Sari Poernomo
  International Journal for Numerical Methods in Engineering, Wiley, 2011, 86 (11), pp.1360-1378. We consider a time-harmonic acoustic scattering problem in a 2D infinite waveguide with walls covered with an absorbing material, in the presence of a mean flow assumed uniform far from the source. To make this problem suitable for a finite element analysis, the infinite domain is truncated. This paper concerns the derivation of a non-reflecting boundary condition on the artificial boundary by means of a Dirichlet-to-Neumann (DtN) map based on a modal decomposition. Compared with the hard-walled guide case, several difficulties are raised by the presence of both the liner and the mean flow. In particular, acoustic modes are no longer orthogonal and behave asymptotically like the modes of a soft-walled guide. However, an accurate approximation of the DtN map can be derived using some bi-orthogonality relations, valid asymptotically for high-order modes. Numerical validations show the efficiency of the method. The influence of the liner with or without mean flow is illustrated. (10.1002/nme.3108)
  DOI : 10.1002/nme.3108
• Time reversal and scattering theory for time-dependent acoustic waves in a homogeneous medium
  
  • Ben-Amar Chokri
  • Hazard Christophe
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2011, 76, pp.938-955.
• Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials
  
  • Chesnel Lucas
  • Ciarlet, Patrick
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2011, 43 (5), pp.2150-2169. In a meta-material, the electric permittivity and/or the magnetic permeability can be negative in given frequency ranges. We investigate the solution of the time-harmonic Maxwell equations in a composite material, made up of classical materials, and meta-materials with negative electric permittivity, in a two-dimensional bounded domain Ω. We study the imbedding of the space of electric fields into L²(Ω)². In particular, we extend the famous result of Weber, proving that it is compact. This result is obtained by studying the regularity of the fields. We first isolate their most singular part, using a decomposition à la Birman-Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to Hˢ(Ω)², for some s > 0. Finally, we show that the compact imbedding result holds as soon as no ratio of permittivities between two adjacent materials is equal to −1. (10.1137/100810903)
  DOI : 10.1137/100810903
• On the use of Lamb modes in the linear sampling method for elastic waveguides
  
  • Bourgeois Laurent
  • Le Louër Frédérique
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2011, 27 (5), pp.055001. This paper is devoted to a modal formulation of the linear sampling method in elastic 2D or 3D waveguide, that is we use the guided modes (called Lamb modes in 2D) as incident waves and the corresponding far fields in order to retrieve some obstacles. We provide the mathematical background to tackle the problem of identifiability and the justification of the linear sampling method for such a case. The elastic waveguide raises a specific issue: it concerns the projection of the scattered field on a transverse basis, which requires the introduction of new variables that mix displacement and stress components and satisfy the so-called Fraser's biorthogonality condition. Some numerical experiments in 2D show the feasibility of the reconstruction in the case of a finite number of incident waves formed by Lamb modes. © 2011 IOP Publishing Ltd. (10.1088/0266-5611/27/5/055001)
  DOI : 10.1088/0266-5611/27/5/055001
• An Error Estimate for the Finite Difference Scheme for One-Phase Obstacle Problem
  
  • Arakelyan Avetik
  • Barkhudaryan Rafayel
  • Poghosyan Michael
  Journal of Contemporary Mathematical Analysis, 2011, 46 (3), pp.131-141.
• Optimal Control of the Atmospheric Reentry of a Space Shuttle by an Homotopy Method
  
  • Hermant Audrey
  Optimal Control Applications and Methods, Wiley, 2011, 32 (6), pp.627-646. This paper deals with the optimal control problem of the atmospheric reentry of a space shuttle with a second-order state constraint on the thermal flux. We solve the problem using the shooting algorithm combined with an homotopy method which automatically determines the structure of the optimal trajectory (composed of one boundary arc and one touch point). (10.1002/oca.961)
  DOI : 10.1002/oca.961
• About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Applicable Analysis, Taylor & Francis, 2011, 90 (10), pp.1481-1497. This article is devoted to a new method to find defects in some elasticplastic structure in the antiplane case from overdetermined data on a subpart of its boundary. This iterative method merges the method of quasireversibility and a simple level set technique. The foundations of the method are justified from a theoretical point of view and its efficiency is shown with the help of numerical experiments. © 2011 Taylor & Francis. (10.1080/00036811.2010.549481)
  DOI : 10.1080/00036811.2010.549481
• Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case
  
  • Barbu Viorel
  • Roeckner Michael
  • Russo Francesco
  Probability Theory and Related Fields, Springer Verlag, 2011. We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution. (10.1007/s00440-010-0291-x)
  DOI : 10.1007/s00440-010-0291-x
• Stable reconstruction of generalized impedance boundary conditions
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2011, 27 (9), pp.095002. (10.1088/0266-5611/27/9/095002)
  DOI : 10.1088/0266-5611/27/9/095002
• Dynamic consistency for Stochastic Optimal Control problems
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • Cohen Guy
  • de Lara Michel
  • Girardeau Pierre
  Annals of Operations Research, Springer Verlag, 2011, pp.17 p.. For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step $t_0$, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time step $t_0, t_1, \dots, T$; at the next time step $t_1$, he is able to formulate a new optimization problem starting at time $t_1$ that yields a new sequence of optimal decision rules. This process can be continued until final time $T$ is reached. A family of optimization problems formulated in this way is said to be time consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of time consistency, well-known in the field of Economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (2007) and studied in the Stochastic Programming framework by Shapiro (2009) and for Markov Decision Processes (MDP) by Ruszczynski (2009). We here link this notion with the concept of ''state variable'' in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen. (10.1007/s10479-011-1027-8)
  DOI : 10.1007/s10479-011-1027-8
• Characterization of the value function of final state constrained control problems with BV trajectories
  
  • Briani Ariela
  • Zidani Hasnaa
  Communications on Pure and Applied Mathematics, Wiley, 2011, 10 (6), pp.1567-1587. This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations. (10.3934/cpaa.2011.10.1567)
  DOI : 10.3934/cpaa.2011.10.1567
• 10e colloque national en calcul des structures (CSMA 2011)
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2011, pp.273 articles. Ce dixième colloque a vocation d'aborder tous les thèmes en rapport avec le calcul des structures : Modélisation et comportement : Structures minces, élasto-visco-plasticité, rupture, fatigue, endommagement, matériaux composites, mise en forme, procédés de fabrication et d'assemblage. Méthodes et techniques de résolution : Parallélisme, sous-structuration, contact, frottement, approches multiéchelles, approches stochastiques, prise en compte des incertitudes, méthodes alternatives aux éléments finis. Contrôle et optimisation : Estimation d'erreur et applications, maillage adaptatif, problèmes inverses, identification, recalage, optimisation de forme et de processus, structures adaptatives, contrôle actif, structures intelligentes. Dynamique, interactions et couplages : Dynamique rapide et vibrations non linéaires, crash, vibroacoustique, interactions multiphysiques, sol-structure, fluide-structure. Applications : Industries mécaniques, biens d'équipement, aéronautique et spatial, véhicules et transport, construction navale, génie civil, énergie, nanotechnologies, biomécanique.
• A discontinuous Galerkin solver for front propagation
  
  • Bokanowski Olivier
  • Cheng Yingda
  • Shu Chi-Wang
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2011, 33 (2), pp.923-938. We propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation.
• Stability of acoustic propagation in 2D duct flows: A low frequency approach
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duruflé Marc
  • Joly Patrick
  • Joubert Lauris
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2011, 21 (5), pp.1121-1151. In this paper, we derive a simplified "quasi-1D" limit model for the propagation of low frequency acoustic waves in a laminar flow filling a 2D duct. We analyze the well-posedness of this model in function of the Mach profile of the flow. This can be reduced to the study of the spectrum of a bounded non-normal operator. As a by-product of this analysis, we establish new results for hydrodynamic instabilities of KelvinHelmholtz type in compressible fluids. © 2011 World Scientific Publishing Company. (10.1142/s0218202511005271)
  DOI : 10.1142/s0218202511005271
• Mathematical and numerical modelling of wave propagation in fractal trees
  
  • Joly Patrick
  • Semin Adrien
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011.
• Application of convex lexicographical optimization to the balance of GRTgaz gas grid
  
  • Bonnans Joseph Frederic
  • Paraisy Ruben
  • Veyrat Sébastien
  • Adam Soizic
  Journal of Global Optimization, Springer Verlag, 2011, 49 (3), pp.415--423.
• An alternative to Dirichlet-to-Neumann maps for waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011, 349 (17-18), pp.1005-1009. We are interested by the treatment of the radiation condition at infinity for the numerical solution of a problem set in an unbounded waveguide. We propose an alternative to the classical approach involving a modal expression of Dirichlet-to-Neumann (DtN) operators. This method is particularly simple to implement since it only requires the solution of boundary value problems with local boundary conditions. The corresponding approximate solution is comparable in accuracy to the one obtained by truncating the infinite series in the DtN maps. (10.1016/j.crma.2011.08.006)
  DOI : 10.1016/j.crma.2011.08.006
• On stochastic calculus related to financial assets without semimartingales
  
  • Coviello Rosanna
  • Di Girolami Cristina
  • Russo Francesco
  Bulletin des Sciences Mathématiques, Elsevier, 2011, 135, pp.733-774. This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class $\mathcal{A}$ of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus related to $\mathcal{A}$-martingales with some examples is developed. Some applications to no-arbitrage, viability, hedging and the maximization of the utility of an insider are expanded. We finally revisit some no arbitrage conditions of Bender-Sottinen-Valkeila type. (10.1016/j.bulsci.2011.06.008)
  DOI : 10.1016/j.bulsci.2011.06.008
• Characterisation of the value function of final state constrained control problems with BV trajectories
  
  • Briani Ariela
  • Zidani Hasnaa
  Communications on Pure and Applied Analysis, AIMS American Institute of Mathematical Sciences, 2011, 10 (6), pp.1567-1587. This paper aims to investigate a control problem governed by differential equations with Random measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo), we obtain that the value function can be characterized by means of an auxiliary control problem involving absolutely continuous trajectories. We study the characterization of the value function of this auxiliary problem and discuss its discrete approximations. (10.3934/cpaa.2011.10.1567)
  DOI : 10.3934/cpaa.2011.10.1567