Partial Differential Equations and Group Theory

New Perspectives for Applications
1. Auflage. Softcover version of original hardcover Auflage 1994
ISBN: 978-90-481-4432-7
Verlag: Springer Netherlands

Buch, Englisch, Band 293, 478 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 739 g

Reihe: Mathematics and Its Applications

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.

Zielgruppe

Research

Autoren/Hrsg.

Pommaret, J. F.

Fachgebiete

• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Technische Thermodynamik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Naturwissenschaften Physik Thermodynamik
• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

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.