Publications

2014

• Optimal feedback control of undamped wave equations by solving a HJB equation
  
  • Kröner Axel
  • Kunisch Karl
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 21 (2), pp.442 - 464. An optimal fi nite-time horizon feedback control problem for (semi linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on nite elements lead to approximated problems governed by ODEs in high dimensional space which makes infeasible the numerical resolution by HJB approach. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The e ffect of noise is considered and numerical simulations are presented to show the relevance of the approach. (10.1051/cocv/2014033)
  DOI : 10.1051/cocv/2014033
• The finite element method in solid mechanics
  
  • Bonnet Marc
  • Frangi Attilio
  • Rey Christian
  , 2014, pp.365. The book focuses on topics that are at the core of the Finite Element Method (FEM) for the mechanics of deformable solids and structures.Its main objective is to provide the reader, who is assumed to be familiar with standard continuum solid mechanics, with a clear grasp of the essentials, sufficient background for reading and exploiting the research literature on computational solid mechanics, and a working knowledge of the main implementational issues of the FEM.This book arises from a course taught since 2004 to last-year students of Ecole Polytechnique (France). It is intended for Master and PhD students, as well as scientists and engineers looking for a rigorous introduction to FEM theory and programming for linear and non-linear analyses in solid mechanics.As a distinguishing feature, in addition to sections devoted to theory and concepts presented in general terms, each chapter also features other sections (interspersed with the former) devoted to detailed description of specific features (e.g. the construction of a specific finite element), annotated Matlab code and/or numerical examples produced with it, or worked-out analytical examples.
• The "exterior approach" to solve the inverse obstacle problem for the Stokes system
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2014, pp.Pages: 23 - 51. We apply an "exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements. Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming infite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand. Some numerical experiments in 2D show the effciency of the two mixed formulations and of the exterior approach based on one of them. (10.3934/ipi.2014.8.23)
  DOI : 10.3934/ipi.2014.8.23
• Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning
  
  • Jean Frédéric
  , 2014. Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems. The notes are divided into three chapters and two appendices. In Chapter 1 we introduce the basic definitions on nonholonomic systems and sub-Riemannian geometry, and give the main result on controllability, namely the Chow-Rashevsky Theorem. Chapter 2 provides a detailed exposition of the notions of first-order approximation, including nonholonomic orders, privileged coordinates, nilpotent approximations, and distance estimates such as the Ball-Box Theorem. As an application we show how these notions allow us to describe the tangent structure to a Carnot-Carathéodory space (the metric space defined by a sub-Riemannian distance). The chapter ends with the presentation of desingularization procedures, that are necessary to recover uniformity in approximations and distance estimates. Chapter 3 is devoted to the motion planning problem for nonholonomic systems. We show in particular how to apply the tools from sub-Riemannian geometry in order to give solutions to this problem, first in the case where the system is nilpotent, and then in the general case. An overview of the existing methods for nonholonomic motion planning concludes this chapter. Finally, we present some results on composition of flows in connection with the Campbell-Hausdorff formula in Appendix A, and some complements on the different systems of privileged coordinates in Appendix B. (10.1007/978-3-319-08690-3)
  DOI : 10.1007/978-3-319-08690-3