Publications

2014

• Non-holonomic interpolation motion planning for the car with trailers
  
  • Gauthier Jean-Paul
  • Monroy-Pérez Felipe
  • Jonathan Laporte
  , 2014.
• Optimization of running strategies based on anaerobic energy and variations of velocity
  
  • Aftalion Amandine
  • Bonnans J. Frederic
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2014, 74 (5), pp.1615-1636. We present new models, numerical simulations and rigorous analysis for the optimization of the velocity in a race. In a seminal paper, Keller (1973,1974) explained how a runner should determine his speed in order to run a given distance in the shortest time. We extend this analysis, based on the equation of motion and aerobic energy, to include a balance of anaerobic energy (or accumulated oxygen deficit) and an energy recreation term when the speed decreases. We also take into account that when the anaerobic energy gets too low, the oxygen uptake cannot be maintained to its maximal value. Using optimal control theory, we obtain a proof of Keller's optimal race, and relate the problem to a relaxed formulation, where the propulsive force represents a probability distribution rather than a value function of time. Our analysis leads us to introduce a bound on the variations of the propulsive force to obtain a more realistic model which displays oscillations of the velocity. Our numerical simulations qualitatively reproduce quite well physiological measurements on real runners. We show how, by optimizing over a period, we recover these oscillations of speed. We point out that our numerical simulations provide in particular the exact instantaneous anaerobic energy used in the exercise.
• Méthodes numériques pour la spatialisation sonore, de la simulation à la synthèse binaurale
  
  • Aussal Matthieu
  , 2014. Les recherches effectuées lors de ce doctorat furent initiées par un besoin industriel identifié par la société Digital Media Solutions, à savoir la réalisation d'un outil de spatialisation sonore au casque destiné aux non-voyants. Malgré les nombreux dispositifs déjà existants, les techniques de son 3D sur casque butent, depuis leur apparition, sur un obstacle majeur, qui empêche la réalisation de produits de haute qualité. En effet, pour rendre l'illusion d'un espace sonore perceptible, il est nécessaire d'individualiser l'écoute pour chaque auditeur. Des mesures en laboratoire offrent des résultats convaincants, mais le processus n'est pas encore applicable au grand public. La modélisation mathématique de cette expérience permet d'envisager de remplacer la mesure par une simulation numérique. Cette thèse propose à la fois un moteur de calcul numérique efficace et adapté au problème posé, ainsi qu'un moteur de rendu audio 3D pour valider cette approche
• Hölder equivalence of the value function for control-affine systems
  
  • Prandi Dario
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (4), pp.1224 - 1248. We prove the continuity and the Hölder equivalence w.r.t.\ an Euclidean distance of the value function associated with the $L^1$ cost of the control-affine system $\dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q)$, satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the $L^1$ cost of time-dependent systems of the form $\dot q = \sum_{j=1}^m u_j f_j^t(q)$. (10.1051/cocv/2014014)
  DOI : 10.1051/cocv/2014014
• Splitting trees with neutral mutations at birth
  
  • Richard Mathieu
  Stochastic Processes and their Applications, Elsevier, 2014, pp.Volume 124, Issue 10, October 2014, Pages 3206-3230. We consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give birth at constant rate to clonal or mutant children in an infinitely many alleles model with neutral mutations. First, to study the allelic partition of the population, we are interested in its frequency spectrum, which, at a fixed time, describes the number of alleles carried by a given number of individuals and with a given age. We compute the expected value of this spectrum and obtain some almost sure convergence results thanks to classical properties of Crump-Mode-Jagers (CMJ) processes counted by random characteristics. Then, by using multitype CMJ-processes, we get asymptotic properties about the number of alleles that have undergone a fixed number of mutations with respect to the ancestral allele of the population. (10.1016/j.spa.2014.05.008)
  DOI : 10.1016/j.spa.2014.05.008
• Mean curvature flow with obstacles: a viscosity approach
  
  • Mercier Gwenael
  , 2014. We introduce a level-set formulation for the mean curvature flow with obstacles and show existence and uniqueness of a viscosity solution. These results generalize a well known viscosity approach for the mean curvature flow without obstacle by Evans and Spruck and Chen, Giga and Goto in 1991. In addition, we show that this evolution is consistent with the variational scheme introduced by Almeida, Chambolle and Novaga (2012) and we study the long time behavior of our viscosity solutions.
• Singular perturbation of optimal control problems on multi-domains
  
  • Forcadel Nicolas
  • Rao Zhiping
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (5), pp.2917–2943. The goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi-domains. We consider an optimal control problem in which the controlled system contains a fast and a slow variables. This problem is reformulated as an Hamilton-Jacobi-Bellman (HJB) equation. The main difficulty comes from the fact that the fast variable lives in a multi-domain. The geometric singularity of the multi-domains leads to the discontinuity of the Hamiltonian. Under a controllability assumption on the fast variables, the limit equation (as the velocity of the fast variable goes to infinity) is obtained via a PDE approache and by means of the tools of the control theory. (10.1137/130916709)
  DOI : 10.1137/130916709
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') -section tridimensionnelle
  
  • Colonna Jean-François
  , 2014. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') -tridimensional cross-section- (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') -section tridimensionnelle-)
• Ramsey Rule with Progressive utility and Long Term Affine Yields Curves
  
  • El Karoui Nicole
  • Mrad Mohamed
  • Hillairet Caroline
  International Journal of Financial Engineering, World Scientific, 2014. he purpose of this paper relies on the study of long term affine yield curves modeling. It is inspired by the Ramsey rule of the economic literature, that links discount rate and marginal utility of aggregate optimal consumption. For such a long maturity modelization, the possibility of adjusting preferences to new economic information is crucial, justifying the use of progressive utility. This paper studies, in a framework with affine factors, the yield curve given from the Ramsey rule. It first characterizes consistent progressive utility of investment and consumption, given the optimal wealth and consumption processes. A special attention is paid to utilities associated with linear optimal processes with respect to their initial conditions, which is for example the case of power progressive utilities. Those utilities are the basis point to construct other progressive utilities generating non linear optimal processes but leading yet to still tractable computations. This is of particular interest to study the impact of initial wealth on yield curves.
• Mean curvature flow with obstacles: existence, uniqueness and regularity of solutions
  
  • Mercier Gwenael
  • Novaga Matteo
  , 2014. We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.
• Electromagnetic inverse shape problem for coated obstacles
  
  • Chaulet Nicolas
  • Haddar Houssem
  Advances in Computational Mathematics, Springer Verlag, 2014, pp.21. We address the inverse problem of retrieving the shape of an obstacle with impedance in the form of a surface wave operator using the knowledge of electromagnetic scattering amplitude at a fixed frequency. We prove unique reconstructions from infinitely many measures. We then provide a characterization of the scattering amplitude derivative with respect to the obstacle shape. This derivative includes the case of shape dependent impedance parameters. We then employ a gradient-descent algorithm with H 1 boundary regularisation of the descent direction to numerically solve the inverse problem. The procedure is validated for three dimensional geometries using synthetic data.
• Une coupe très intéressante dans l'éponge de Menger -itération 5
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the Menger sponge -iteration 5- (Une coupe très intéressante dans l'éponge de Menger -itération 5-)
• L'éponge de Menger -itération 5
  
  • Colonna Jean-François
  , 2014. The Menger sponge -iteration 5- (L'éponge de Menger -itération 5-)
• Une coupe très intéressante dans le complément de l'éponge de Menger -itération 4
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the complement of the Menger sponge -iteration 4- (Une coupe très intéressante dans le complément de l'éponge de Menger -itération 4-)
• Une coupe très intéressante dans le complément de l'éponge de Menger -itération 3
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the complement of the Menger sponge -iteration 3- (Une coupe très intéressante dans le complément de l'éponge de Menger -itération 3-)
• Une coupe très intéressante dans le complément de l'éponge de Menger -itération 1
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the complement of the Menger sponge -iteration 1- (Une coupe très intéressante dans le complément de l'éponge de Menger -itération 1-)
• Une coupe très intéressante dans le complément de l'éponge de Menger -itération 2
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the complement of the Menger sponge -iteration 2- (Une coupe très intéressante dans le complément de l'éponge de Menger -itération 2-)
• Le complément de l'éponge de Menger -itération 3
  
  • Colonna Jean-François
  , 2014. The complement of the Menger sponge -iteration 3- (Le complément de l'éponge de Menger -itération 3-)
• Le complément de l'éponge de Menger -itération 4
  
  • Colonna Jean-François
  , 2014. The complement of the Menger sponge -iteration 4- (Le complément de l'éponge de Menger -itération 4-)
• L'éponge de Menger -itération 4
  
  • Colonna Jean-François
  , 2014. The Menger sponge -iteration 4- (L'éponge de Menger -itération 4-)
• L'éponge de Menger -itération 1
  
  • Colonna Jean-François
  , 2014. The Menger sponge -iteration 1- (L'éponge de Menger -itération 1-)
• Le complément de l'éponge de Menger -itération 2
  
  • Colonna Jean-François
  , 2014. The complement of the Menger sponge -iteration 2- (Le complément de l'éponge de Menger -itération 2-)
• Le complément de l'éponge de Menger -itération 1
  
  • Colonna Jean-François
  , 2014. The complement of the Menger sponge -iteration 1- (Le complément de l'éponge de Menger -itération 1-)
• Une coupe très intéressante dans l'éponge de Menger -itération 4
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the Menger sponge -iteration 4- (Une coupe très intéressante dans l'éponge de Menger -itération 4-)
• Une coupe très intéressante dans l'éponge de Menger -itération 3
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the Menger sponge -iteration 3- (Une coupe très intéressante dans l'éponge de Menger -itération 3-)