Partial Differential Equations and Group Theory: New Perspectives for Applications / Edition 1

by J.F. Pommaret

Overview

Ordinary differential control thPory (the classical theory) studies input/output relations defined by systems of ordinary differential equations (ODE). The various concepts 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 concepts 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 differential elimination presented here will allow us to determine compatibility conditions on input and/or output as a better understanding of the foundations of control theory. At the same time we shall unify differential geometry and differential algebra in a new framework, called differential algebraic geometry.

Product Details

ISBN-13:	9789048144327
Publisher:	Springer Netherlands
Publication date:	12/07/2010
Series:	Mathematics and Its Applications , #293
Edition description:	Softcover reprint of hardcover 1st ed. 1994
Pages:	478
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

I. Homological Algebra.- II. Jet Theory.- III. Nonlinear Systems.- IV. Linear Systems.- V. Group Theory.- VI. Differential Galois Theory.- VII. Control Theory.- VIII. Continuum Physics.