Wave Propagation, Observation and Control in 1-d Flexible Multi-structures

1 Introduction

2 Preliminaries

2.1 The Elastic String

2.2 Networks of Strings

2.2.1 Elements on Graphs

2.2.2 Equations of Motion for Networks

2.3 The Control Problem

2.3.1 Basic Definitions

2.3.2 An Equivalent Formulation of the Control Problem

2.4 A Controllability Theorem and its Limitations

3 SomeUsefulTools

3.1 D'Alembert Formula and Boundary Observability of the 1 - d Wave Equation

3.1.1 D'Alembert Formula

3.1.2 Boundary Observability of the 1 - d Wave Equation

3.2 HUM

3.2.1 Description of the Method

3.2.2 Application to the Control of Networks

3.3 The Method of Moments

3.3.1 Description of the Method

3.3.2 Application to the Control of Networks

3.4 Riesz Bases and Ingham-Type Inequalities

3.4.1 Riesz Bases

3.4.2 Generalized Ingham Theorems

3.4.3 A New Inequality

4 The Three String Network

4.1 The Three String Network with Two Controlled Nodes

4.1.1 Equations of Motion of the Network

4.1.2 The Control Problem

4.2 A Simpler Problem: Simultaneous Control of Two Strings

4.2.1 Identification of Controllable Subspaces

4.3 The Three String Network with One Controlled Node

4.4 An Observability Inequality

4.5 Properties of the Sequence of Eigenvalues

4.6 Observability of the Fourier Coefficients

4.7 Study of the Weights c_n 

4.8 Relation with the Simultaneous Control of Two Strings

4.9 Lack of Observability in Small Time

4.10 Application of the Method of Moments to Control

5 General Trees

5.1 Notations and Statement of the Problem

5.1.1 Notations for Graphs

5.1.2 Equations of Motion

5.2 The Operators P and Q

5.2.1 A Tree Formed by a Single String

5.2.2 Operators of Type S

5.2.3 Construction of P and Q in the General Case

5.2.4 The Action of P and Q at the Interior Nodes

5.2.5 Action of P and Q on the Solution

5.3 The Main Observability Result

5.4 Relation Between P and Q and the Spectrum

5.4.1 The Eigenvalue Problem

5.4.2 Further Properties of P and Q

5.5 Observability Results

5.5.1 Weighted Observability Inequalities

5.5.2 Non-degenerate Trees

5.5.3 On the Set of Non-degenerate Trees

5.6 Consequences Concerning Controllability

5.7 Simultaneous Observability and Controllability of Networks

5.8 Examples

5.8.1 The Star-Shaped Network with n Strings

5.8.2 Simultaneous Control of n Strings

5.8.3 A Non Star-Shaped Tree

6 Some Observability and Controllability Results for General Networks

6.1 Spectral Control of General Networks

6.1.1 Asymptotic Behavior of the Eigenfunctions

6.1.2 Application to Control

6.2 Colored Networks

6.3 Optimality of Theorem 3.2.7

6.3.1 Simultaneous Control of Serially Connected Strings

7 Simultaneous Observation and Control from an Interior Region

7.1 Simultaneous Interior Control of Two Strings

7.1.1 Statement of the Problem

7.1.2 Control of Strings with Different Densities

7.1.3 Control of Strings with Equal Densities

7.2 Simultaneous Control on the Whole Domain

8 Other Equations on Networks

8.1 The Heat Equation

8.2 The Schrödinger Equation

8.3 A Model of Network for Beams

9 Final Remarks and Open Problems

9.1 Brief Description of the Main Results of the Book

9.1.1 Networks of Strings

9.1.2 Simultaneous Control of Strings

9.1.3 Other Equations on Networks

9.2 Future Lines of Research and Open Problems

Some Consequences of Diophantine Approximation Theorems

References

Index