Advertisement

The Convergence Problem for Dissipative Autonomous Systems

Classical Methods and Recent Advances

  • Alain Haraux
  • Mohamed Ali Jendoubi

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Alain Haraux, Mohamed Ali Jendoubi
    Pages 1-3
  3. Alain Haraux, Mohamed Ali Jendoubi
    Pages 5-17
  4. Alain Haraux, Mohamed Ali Jendoubi
    Pages 19-28
  5. Alain Haraux, Mohamed Ali Jendoubi
    Pages 29-35
  6. Alain Haraux, Mohamed Ali Jendoubi
    Pages 37-44
  7. Alain Haraux, Mohamed Ali Jendoubi
    Pages 45-65
  8. Alain Haraux, Mohamed Ali Jendoubi
    Pages 67-76
  9. Alain Haraux, Mohamed Ali Jendoubi
    Pages 77-90
  10. Alain Haraux, Mohamed Ali Jendoubi
    Pages 91-99
  11. Alain Haraux, Mohamed Ali Jendoubi
    Pages 101-114
  12. Alain Haraux, Mohamed Ali Jendoubi
    Pages 115-132
  13. Alain Haraux, Mohamed Ali Jendoubi
    Pages 133-139
  14. Back Matter
    Pages 141-142

About this book

Introduction

The book investigates classical and more recent methods of study for the asymptotic behavior of dissipative continuous dynamical systems with applications to ordinary and partial differential equations, the main question being convergence (or not) of the solutions to an equilibrium. After reviewing the basic concepts of topological dynamics and the definition of gradient-like systems on a metric space, the authors present a comprehensive exposition of stability theory relying on the so-called linearization method. For the convergence problem itself, when the set of equilibria is infinite, the only general results that do not require very special features of the non-linearities are presently consequences of a gradient inequality discovered by S. Lojasiewicz. The application of this inequality jointly with the so-called Liapunov-Schmidt reduction requires a rigorous exposition of Semi-Fredholm operator theory and the theory of real analytic maps on infinite dimensional Banach spaces, which cannot be found anywhere in a readily applicable form. The applications covered in this short text are the simplest, but more complicated cases are mentioned in the final chapter, together with references to the corresponding specialized papers.

Keywords

Dynamical systems Gradient inequality PDE Stability of equilibria Trend to equilibrium

Authors and affiliations

  • Alain Haraux
    • 1
  • Mohamed Ali Jendoubi
    • 2
  1. 1.Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598Laboratoire Jacques-Louis LionsParis France
  2. 2.Université de CarthageInstitut Préparatoire aux Etudes Scientifiques et TechniquesLa MarsaTunisia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-23407-6
  • Copyright Information The Author(s) 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-23406-9
  • Online ISBN 978-3-319-23407-6
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site
Industry Sectors
Aerospace