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Partial Differential Equations and Group Theory

New Perspectives for Applications

  • J.-F. Pommaret

Part of the Mathematics and Its Applications book series (MAIA, volume 293)

Table of contents

  1. Front Matter
    Pages i-ix
  2. J.-F. Pommaret
    Pages 1-47
  3. J.-F. Pommaret
    Pages 49-60
  4. J.-F. Pommaret
    Pages 61-80
  5. J.-F. Pommaret
    Pages 81-137
  6. J.-F. Pommaret
    Pages 139-175
  7. J.-F. Pommaret
    Pages 177-258
  8. J.-F. Pommaret
    Pages 259-318
  9. J.-F. Pommaret
    Pages 319-389
  10. J.-F. Pommaret
    Pages 391-456
  11. Back Matter
    Pages 457-477

About this book

Introduction

Ordinary differential control thPory (the classical theory) studies input/output re­ lations defined by systems of ordinary differential equations (ODE). The various con­ cepts that can be introduced (controllability, observability, invertibility, etc. ) must be tested on formal objects (matrices, vector fields, etc. ) by means of formal operations (multiplication, bracket, rank, etc. ), but without appealing to the explicit integration (search for trajectories, etc. ) of the given ODE. Many partial results have been re­ cently unified by means of new formal methods coming from differential geometry and differential algebra. However, certain problems (invariance, equivalence, linearization, etc. ) naturally lead to systems of partial differential equations (PDE). More generally, partial differential control theory studies input/output relations defined by systems of PDE (mechanics, thermodynamics, hydrodynamics, plasma physics, robotics, etc. ). One of the aims of this book is to extend the preceding con­ cepts to this new situation, where, of course, functional analysis and/or a dynamical system approach cannot be used. A link will be exhibited between this domain of applied mathematics and the famous 'Backlund problem', existing in the study of solitary waves or solitons. In particular, we shall show how the methods of differ­ ential elimination presented here will allow us to determine compatibility conditions on input and/or output as a better understanding of the foundations of control the­ ory. At the same time we shall unify differential geometry and differential algebra in a new framework, called differential algebraic geometry.

Keywords

Group theory Nonlinear system algorithms computer algebra mathematical physics partial differential equation

Authors and affiliations

  • J.-F. Pommaret
    • 1
  1. 1.Centre de Recherche en Mathématiques Appliquées (CERMA)Ecole Nationale des Ponts et Chausées (ENPC)Noisy-le-GrandFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-2539-2
  • Copyright Information Springer Science+Business Media B.V. 1994
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4432-7
  • Online ISBN 978-94-017-2539-2
  • Buy this book on publisher's site
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