Fourier Series in Control Theory

This book uses techniques of Fourier series and functional analysis to deal with certain problems in differential equations. The Fourier series and functional analysis are merely tools; the authors' real interest lies in the differential equations that they study. It has been known since 1967 that a wide variety of sets {e wikt } of complex exponential functions play an important role in the control theory of systems governed by partial differential equations. However, this book is the first serious attempt to gather all of the available theory of these "nonharmonic Fourier series" in one place, combining published results with new results by the authors, to create a unique source of such material for practicing applied mathematicians, engineers and other scientific professionals.

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

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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.

Résumé

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.

Contenu
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.