Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures (eBook)

Preface 6
Contents 8
1 Introduction 11
2 Preliminaries 18
2.1 The Elastic String 18
2.2 Networks of Strings 23
2.3 The Control Problem 30
2.4 A Controllability Theorem and its Limitations 33
3 Some Useful Tools 34
3.1 D’Alembert Formula and Boundary Observability 34
of the 1 34
Wave Equation 34
3.2 The Hilbert Uniqueness Method (HUM): Reduction 37
to an Observability Problem. 37
3.3 The Method of Moments 45
3.4 Riesz Bases and Ingham-Type Inequalities 54
4 The Three String Network 62
4.1 The Three String Network with Two Controlled 62
Nodes 62
4.2 A Simpler Problem: Simultaneous Control of Two 66
Strings 66
Positive results 72
Negative results 73
4.3 The Three String Network with One Controlled 76
Node 76
4.4 An Observability Inequality 78
4.5 Properties of the Sequence of Eigenvalues 85
4.6 Fourier Representation of the Observability 89
Inequality 89
4.7 Study of the Weights 90
4.8 Relation Between the Simultaneous Control of Two 96
Strings and the Control of the Three String Network 96
from One Exterior Node 96
4.9 Lack of Observability in Small Time 99
4.10 Application of the Method of Moments 105
to the Control of the Three String Network 105
5 General Trees 111
5.1 Notations and Statement of the Problem 112
5.2 The Operators 116
and 116
5.3 The Main Observability Result 127
5.4 Relation Between 131
and 131
and the Spectrum 131
5.5 Observability Results 139
5.6 Consequences Concerning Controllability 145
5.7 Simultaneous Observability and Controllability 146
of Networks 146
5.8 Examples 150
6 Some Observability and Controllability Results for General Networks 157
6.1 Spectral Control of General Networks 158
6.2 Colored Networks 165
6.3 Optimality of Theorem 3.2.7 169
7 Simultaneous Observation and Control from an Interior Region 175
7.1 Simultaneous Interior Control of Two Strings 176
7.2 Simultaneous Control on the Whole Domain 185
8 Other Equations on Networks 190
8.1 The Heat Equation 191
8.2 The Schr¨ odinger Equation 194
8.3 A Model of Network for Beams 199
9 Final Remarks and Open Problems 203
9.1 Brief Description of the Main Results of the Book 203
9.2 Future Lines of Research and Open Problems 205
A Some Consequences of Diophantine Approximation Theorems 210
References 217
Index 224