Partial Differential Equations and Group Theory

1. Front Matter

   Pages i-ix

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2. Introduction

   • J.-F. Pommaret

   Pages 1-47

3. Homological Algebra

   • J.-F. Pommaret

   Pages 49-60

4. Jet Theory

   • J.-F. Pommaret

   Pages 61-80

5. Nonlinear Systems

   • J.-F. Pommaret

   Pages 81-137

6. Linear Systems

   • J.-F. Pommaret

   Pages 139-175

7. Group theory

   • J.-F. Pommaret

   Pages 177-258

8. Differential Galois Theory

   • J.-F. Pommaret

   Pages 259-318

9. Control Theory

   • J.-F. Pommaret

   Pages 319-389

10. Continuum Physics

    • J.-F. Pommaret

    Pages 391-456

11. Back Matter

    Pages 457-477

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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.

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

• Centre de Recherche en Mathématiques Appliquées (CERMA), Ecole Nationale des Ponts et Chausées (ENPC), Noisy-le-Grand, France

  J.-F. Pommaret

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