Partial Differential Control Theory: Volume I: Mathematical Tools, Volume II: Control System: 530 (Mathematics and Its Applications, 530) - Hardcover

Pommaret, J.F.

Hardcover

ISBN 10: 0792370376 ISBN 13: 9780792370376

Publisher: Springer, 2001

Synopsis

The mathematical theory of "open" dynamical systems is a creation of the twentieth century. Its humble beginnings focused on ideas of Laplace transforms applied to linear problems of automatic control and to the analysis and synthesis of electrical circuits. However during the second half of the century, it flowered into a field based on an array of sophisticated mathematical concepts and techniques from algebra, nonlinear analysis and differential geometry. The central notion is that of a dynamical system that exchanges matter, energy, or information with its surroundings, i.e. an "open" dynamical system. The mathema­ tization of this notion evolved considerably over the years. The early development centered around the input/output point of view and led to important results, particularly in controller design. Thinking about open systems as a "black box" that accepts stimuli and produces responses has had a wide influence also in areas outside engineering, for example in biology, psychology, and economics. In the early 1960's, especially through the work of Kalman, input/state/output models came in vogue. This model class accommodates very nicely the internal initial conditions that are essentially always present in a dynamical system. The introduction of input/state/output models led to a tempestuous development that made systems and control into a mature discipline with a wide range of concepts, results, algorithms, and applications.

About this title

Review

`...the book provides interesting and engaging reading, particularly to those who wish to learn about the algebraic-geometric aspects of control theory.'
Mathematical Reviews, 2002

Synopsis

Algebraic analysis, that is the algebraic study of systems of partial differential equations by means of module theory and homological algebra, was pioneered around 1970 by M. Kashiwara, B. Malgrange, and V.P. Palamodov. The theory of differential modules, namely modules over a noncommutative ring of differential operators, is a fashionable subject of research today. However, despite its fundamental importance in mathematics, it can only be found in specialist books and papers, and has only been applied in control theory since 1990. This book provides a self-contained and exhaustive account of algebraic analysis and its application to control systems defined by partial differential equations. The first volume presents the mathematical tools needed from both commutative algebra, homological algebra, differential geometry and differential algebra. The second volume applies these new methods in order to study the structural and input/output properties of both linear and nonlinear control systems. Hundreds of explicit examples allow the reader to gain insight and experience in these topics.

The book is written at a graduate level and is intended for researchers in mathematics, mathematical physics, computer algebra, control theory, and theoretical mechanics

Edition details

• Publisher: Springer
• Publication date: 2001
• ISBN 10: 0792370376
• ISBN 13: 9780792370376
• Binding: Hardcover
• Language: English
• Number of pages: 965