Fourier Series in Control Theory

Advance Praise for Fourier Series in Control Theory:

It has been known since 1967 that a wide variety of sets of complex exponential functions play an important role in the control theory of systems governed by partial differential equations. Up until now, the relevant literature has been scattered among various journal articles, a survey paper by myself in SIAM Review in 1978, and the book, Families of Exponentials, by S. Avdonin and S. Ivanov (1995). Fourier Series in Control Theory successfully gathers all of the available theory of these "nonharmonic Fourier series" in one place, combining published results with new results, to create a unique source of such material for practicing applied mathematicians, engineers, and other scientific professionals.

- David Russell, Virginia Polytechnic Institute and State University

This monograph will be of interest to experts and researchers, as well as graduate students in such courses as control series and harmonic analysis. Starting with an overview of the problems of observability, controllability, and stabilization of linear systems and their interconnections, the text contains complete proofs along with a short, simplified, presentation of some properties of Bessel functions for the convenience of the reader. Only basic knowledge of functional analysis is required.

List of contents

Observation, Control, and Stabilization.- Well-Posedness in a Riesz Basis Setting.- Observability of Strings.- Observability of Beams.- Vector Sum Estimates.- Problems on Spherical Domains.- Multidimensional Ingham-Type Theorems.- A General Ingham-Type Theorem.- Problems with Weakened Gap Conditions.

Summary

This book is the first serious attempt to gather all of the available theory of  "nonharmonic Fourier series" in one place, combining published results with new results by the authors.

 

Additional text

From the reviews:

"Fourier Series in Control Theory represents the first appearance of a synthesis of the relevant literature as well as a collection of new results. The story begins with the solution of a problem with a vibrating string having two free endpoints. … This is primarily a monograph aimed at experts and researchers in the field, although it would be accessible to a graduate student willing to fill the background. The exposition is clear … ." (William J. Satzer, MathDL, May, 2005)

"This monograph will be of interest to experts and researchers, as well as graduate students in such courses as control series and harmonic analysis. Starting with an overview of the problems of observability, controllability, and stabilization of linear systems and their interconnections, the text contains complete proofs along with a short, simplified, presentation of some properties of Bessel functions for the convenience of the reader. Only basic knowledge of functional analysis is required." (Bulletin Bibliographique, Vol. 51 (1-2), 2005)

"This book is concerned with harmonic (and nonharmonic) analysis methods in control theory. In the reviewer’s opinion, it can be very useful to understand some of the key ideas. These methods can also be applied to analyze the controllability of many time-reversible systems of practical interest. … The book is very well written and can be easily understood by a non-specialist. Only elementary knowledge of functional analysis and partial differential equations theory is required." (Enrique Fernández Cara, Mathematical Reviews, Issue 2006 a)

"This outstanding book discusses new theoretical approaches to the study of control problems based on harmonic analysis. … This volume is primarily addressed to applied mathematicians working in the field of control theory and harmonic analysis. However, the book will also be useful for scientists from the application areas, in particular, appliedscientists from engineering and physics." (Jan Lovíšek, Zentralblatt MATH, Vol. 1094 (20), 2006)

Report

From the reviews:

"Fourier Series in Control Theory represents the first appearance of a synthesis of the relevant literature as well as a collection of new results. The story begins with the solution of a problem with a vibrating string having two free endpoints. ... This is primarily a monograph aimed at experts and researchers in the field, although it would be accessible to a graduate student willing to fill the background. The exposition is clear ... ." (William J. Satzer, MathDL, May, 2005)
"This monograph will be of interest to experts and researchers, as well as graduate students in such courses as control series and harmonic analysis. Starting with an overview of the problems of observability, controllability, and stabilization of linear systems and their interconnections, the text contains complete proofs along with a short, simplified, presentation of some properties of Bessel functions for the convenience of the reader. Only basic knowledge of functional analysis is required." (Bulletin Bibliographique, Vol. 51 (1-2), 2005)
"This book is concerned with harmonic (and nonharmonic) analysis methods in control theory. In the reviewer's opinion, it can be very useful to understand some of the key ideas. These methods can also be applied to analyze the controllability of many time-reversible systems of practical interest. ... The book is very well written and can be easily understood by a non-specialist. Only elementary knowledge of functional analysis and partial differential equations theory is required." (Enrique Fernández Cara, Mathematical Reviews, Issue 2006 a)
"This outstanding book discusses new theoretical approaches to the study of control problems based on harmonic analysis. ... This volume is primarily addressed to applied mathematicians working in the field of control theory and harmonic analysis. However, the book will also be useful for scientists from the application areas, in particular, appliedscientists from engineering and physics." (Jan Lovísek, Zentralblatt MATH, Vol. 1094 (20), 2006)

Product details

• Authors: Vilmo Komornik, Vilmos Komornik, Paola Loreti
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 27.10.2010
• EAN: 9781441919755
• ISBN: 978-1-4419-1975-5
• No. of pages: 226
• Dimensions: 157 mm x 14 mm x 235 mm
• Weight: 363 g
• Illustrations: IX, 226 p. 29 illus.
• Series: Springer Monographs in Mathematics; Springer Monographs in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Miscellaneous; B, Optimization, Mathematics and Statistics, Systems Theory, Control, Calculus of Variations and Optimization, Calculus of variations, Calculus of Variations and Optimal Control; Optimization, System Theory