Numerical Approximation of Exact Controls for Waves

  • Sylvain Ervedoza
  • Enrique Zuazua

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Sylvain Ervedoza, Enrique Zuazua
    Pages 1-48
  3. Sylvain Ervedoza, Enrique Zuazua
    Pages 49-58
  4. Sylvain Ervedoza, Enrique Zuazua
    Pages 79-114
  5. Sylvain Ervedoza, Enrique Zuazua
    Pages 115-118
  6. Back Matter
    Pages 119-122

About this book

Introduction

​​​​​​This book is devoted to fully developing and comparing the two main approaches to the numerical approximation of controls for wave propagation phenomena: the continuous and the discrete. This is accomplished in the abstract functional setting of conservative semigroups.The main results of the work unify, to a large extent, these two approaches, which yield similaralgorithms and convergence rates. The discrete approach, however, gives not only efficient numerical approximations of the continuous controls, but also ensures some partial controllability properties of the finite-dimensional approximated dynamics. Moreover, it has the advantage of leading to iterative approximation processes that converge without a limiting threshold in the number of iterations. Such a threshold, which is hard to compute and estimate in practice, is a drawback of the methods emanating from the continuous approach. To complement this theory, the book provides convergence results for the discrete wave equation when discretized using finite differences and proves the convergence of the discrete wave equation with non-homogeneous Dirichlet conditions. The first book to explore these topics in depth, "On the Numerical Approximations of Controls for Waves" has rich applications to data assimilation problems and will be of interest to researchers who deal with wave approximations.​

Keywords

controllability convergence results finite differences numerical approximation observability wave equation

Authors and affiliations

  • Sylvain Ervedoza
    • 1
  • Enrique Zuazua
    • 2
  1. 1.Université Paul Sabatier & CNRS, Equipe MIPInstitut de Mathématique de ToulouseToulouse Cedex 9France
  2. 2.BCAM-Basque Center for Applied MathematiBilbaoSpain

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-5808-1
  • Copyright Information Sylvain Ervedoza and Enrique Zuazua 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-5807-4
  • Online ISBN 978-1-4614-5808-1
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • About this book