Wave Propagation, Observation and Control in 1 - D Flexible Multi-Structures

by Dager, Rene; Zuazua, Enrique

• ISBN13: 9783540272397
• ISBN10: 3540272399
• Format: Paperback
• Copyright: 2005-11-15
• Publisher: Springer Verlag

Summary

This volume presents a detailed study of partial differential equations on planar graphs modeling networked flexible mechanical structures. Special emphasis is laid on the understanding of wave propagation phenomena, through the analysis of the problems of observability and controllability from small regions of the graph or its boundary. Some of these results are extended to the heat, beam and Schrödinger equations on planar graphs. Designed as a self-contained introductory course on control and observation of networks, the volume contains also some advanced topics and new techniques which may be of interest for researchers in this area. It also includes a list of open problems and topics for future research.

Table of Contents

Introduction	p. 1
Preliminaries	p. 9
The Elastic String	p. 9
Networks of Strings	p. 14
Elements on Graphs	p. 14
Equations of Motion for Networks	p. 15
The Control Problem	p. 21
Basic Definitions	p. 21
An Equivalent Formulation of the Control Problem	p. 22
A Controllability Theorem and its Limitations	p. 24
Some Useful Tools	p. 25
D'Alembert Formula and Boundary Observability of the 1 - d Wave Equation	p. 25
D'Alembert Formula	p. 25
Boundary Observability of the 1 - d Wave Equation	p. 27
HUM	p. 28
Description of the Method	p. 28
Application to the Control of Networks	p. 33
The Method of Moments	p. 36
Description of the Method	p. 36
Application to the Control of Networks	p. 40
Riesz Bases and Ingham-Type Inequalities	p. 45
Riesz Bases	p. 45
Generalized Ingham Theorems	p. 45
A New Inequality	p. 49
The Three String Network	p. 53
The Three String Network with Two Controlled Nodes	p. 53
Equations of Motion of the Network	p. 53
The Control Problem	p. 55
A Simpler Problem: Simultaneous Control of Two Strings	p. 57
Identification of Controllable Subspaces	p. 61
The Three String Network with One Controlled Node	p. 67
An Observability Inequality	p. 69
Properties of the Sequence of Eigenvalues	p. 76
Observability of the Fourier Coefficients	p. 80
Study of the Weights cn	p. 81
Relation with the Simultaneous Control of Two Strings	p. 87
Lack of Observability in Small Time	p. 90
Application of the Method of Moments to Control	p. 96
General Trees	p. 103
Notations and Statement of the Problem	p. 104
Notations for Graphs	p. 104
Equations of Motion	p. 105
The Operators P and Q	p. 108
A Tree Formed by a Single String	p. 108
Operators of Type S	p. 108
Construction of P and Q in the General Case	p. 114
The Action of P and Q at the Interior Nodes	p. 116
Action of P and Q on the Solution	p. 118
The Main Observability Result	p. 119
Relation Between P and Q and the Spectrum	p. 123
The Eigenvalue Problem	p. 124
Further Properties of P and Q	p. 129
Observability Results	p. 131
Weighted Observability Inequalities	p. 131
Non-degenerate Trees	p. 133
On the Set of Non-degenerate Trees	p. 135
Consequences Concerning Controllability	p. 137
Simultaneous Observability and Controllability of Networks	p. 138
Examples	p. 142
The Star-Shaped Network with n Strings	p. 142
Simultaneous Control of n Strings	p. 144
A Non Star-Shaped Tree	p. 146
Some Observability and Controllability Results for General Networks	p. 149
Spectral Control of General Networks	p. 150
Asymptotic Behavior of the Eigenfunctions	p. 150
Application to Control	p. 154
Colored Networks	p. 157
Optimality of Theorem 3.2.7	p. 161
Simultaneous Control of Serially Connected Strings	p. 162
Simultaneous Observation and Control from an Interior Region	p. 167
Simultaneous Interior Control of Two Strings	p. 168
Statement of the Problem	p. 168
Control of Strings with Different Densities	p. 170
Control of Strings with Equal Densities	p. 174
Simultaneous Control on the Whole Domain	p. 177
Other Equations on Networks	p. 183
The Heat Equation	p. 184
The Schrödinger Equation	p. 187
A Model of Network for Beams	p. 192
Final Remarks and Open Problems	p. 197
Brief Description of the Main Results of the Book	p. 197
Networks of Strings	p. 197
Simultaneous Control of Strings	p. 198
Other Equations on Networks	p. 199
Future Lines of Research and Open Problems	p. 199
Some Consequences of Diophantine Approximation Theorems	p. 205
References	p. 213
Index	p. 221
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