Stability and Stabilization of Infinite Dimensional Systems with Applications

Overview

This book reports on recent achievements in stability and feedback stabilization of infinite systems. In particular emphasis is placed on second order partial differential equations, such as Euler-Bernoulli beam equations, which arise from vibration control of flexible robot arms and large space structures. Various control methods such as sensor feedback control and dynamic boundary control are applied to stabilize the equations. Many new theorems and methods are included in the book. Proof procedures of existing...

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Paperback (Softcover reprint of the original 1st ed. 1999)
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Overview

This book reports on recent achievements in stability and feedback stabilization of infinite systems. In particular emphasis is placed on second order partial differential equations, such as Euler-Bernoulli beam equations, which arise from vibration control of flexible robot arms and large space structures. Various control methods such as sensor feedback control and dynamic boundary control are applied to stabilize the equations. Many new theorems and methods are included in the book. Proof procedures of existing theorems are simplified, and detailed proofs have been given to most theorems. In addition to benefiting from the presentation of new results on semigroups and their stability, readers can also learn several useful techniques for solving practical engineering problems. Until now, the recently obtained research results included in this book were unavailable in one volume. This self-contained book is an invaluable source of information for all those who are familiar with some basic theorems of functional analysis.

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Product Details

  • ISBN-13: 9781447111368
  • Publisher: Springer London
  • Publication date: 1/28/2013
  • Series: Communications and Control Engineering Series
  • Edition description: Softcover reprint of the original 1st ed. 1999
  • Edition number: 1
  • Pages: 403
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.86 (d)

Table of Contents

1 Introduction.- 1.1 Overview and examples of infinite dimensional systems.- 1.2 Organization and brief summary.- 1.3 Remarks on notation.- 1.4 Notes and references.- 2 Semigroups of Linear Operators.- 2.1 Motivation and definitions.- 2.2 Properties of semigroups.- 2.3 Generation theorems for semigroups.- 2.4 Relation with the Laplace transform.- 2.5 Differentiability and analytic semigroups.- 2.6 Compact semigroups.- 2.7 Abstract Cauchy problem.- 2.7.1 Homogeneous initial value problems.- 2.7.2 Inhomogeneous initial value problems.- 2.7.3 Lipschitz perturbations.- 2.8 Integrated semigroups.- 2.9 Nonlinear semigroups of contractions.- 2.10 Notes and references.- 3 Stability of C0-Semigroups.- 3.1 Spectral mapping theorems.- 3.2 Spectrum-determined growth condition.- 3.3 Weak stability and asymptotic stability.- 3.4 Exponential stability — time domain criteria.- 3.5 Exponential stability — frequency domain criteria.- 3.6 Essential spectrum and compact perturbations.- 3.7 Invariance principle for nonlinear semigroups.- 3.8 Notes and references.- 4 Static Sensor Feedback Stabilization of Euler-Bernoulli Beam Equations.- 4.1 Modeling of a rotating beam with a rigid tip body.- 4.2 Stabilization using strain or shear force feedback.- 4.3 Damped second order systems.- 4.4 Exponential stability and spectral analysis.- 4.4.1 Exponential stability.- 4.4.2 Spectral analysis.- 4.5 Shear force feedback control of a rotating beam.- 4.5.1 Well-posedness and exponential stability.- 4.5.2 Asymptotic behavior of the spectrum.- 4.6 Stability analysis of a hybrid system.- 4.6.1 Well-posedness and exponential stability.- 4.6.2 Spectral analysis.- 4.7 Gain adaptive strain feedback control of Euler-Bernoulli beams.- 4.8 Notes and references.- 5 Dynamic Boundary Control of Vibration Systems Based on Passivity.- 5.1 A general framework for system passivity.- 5.1.1 Uncontrolled case.- 5.1.2 Controlled case.- 5.2 Dynamic boundary control using positive real controllers.- 5.2.1 Positive real controllers and their characterizations.- 5.2.2 Stability analysis of control systems with SPR controllers.- 5.3 Dynamic boundary control of a rotating flexible beam.- 5.3.1 Stabilization problem using SPR controllers.- 5.3.2 Orientation problem using positive real controllers.- 5.4 Stability robustness against small time delays.- 5.5 Notes and references.- 6 Other Applications.- 6.1 A General linear hyperbolic system.- 6.2 Stabilization of serially connected vibrating strings.- 6.3 Two coupled vibrating strings.- 6.4 A vibration cable with a tip mass.- 6.5 Thermoelastic system with Dirichlet — Dirichlet boundary conditions.- 6.6 Thermoelastic system with Dirichlet — Neumann boundary conditions.- 6.7 Renardy’s counter-example on spectrum-determined growth condition.- 6.8 Notes and references.

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