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

1. Front Matter

   Pages I-IX

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2. Introduction

   Pages 1-7

3. Preliminaries

   Pages 9-24

4. Some Useful Tools

   Pages 25-52

5. The Three String Network

   Pages 53-101

6. General Trees

   Pages 103-148

7. Some Observability and Controllability Results for General Networks

   Pages 149-166

8. Simultaneous Observation and Control from an Interior Region

   Pages 167-181

9. Other Equations on Networks

   Pages 183-195

10. Final Remarks and Open Problems

    Pages 197-203

11. Back Matter

    Pages 205-224

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This book is devoted to analyze the vibrations of simpli?ed 1? d models of multi-body structures consisting of a ?nite number of ?exible strings d- tributed along planar graphs. We?rstdiscussissueson existence and uniquenessof solutions that can be solved by standard methods (energy arguments, semigroup theory, separation ofvariables,transposition,...).Thenweanalyzehowsolutionspropagatealong the graph as the time evolves, addressing the problem of the observation of waves. Roughly, the question of observability can be formulated as follows: Can we obtain complete information on the vibrations by making measu- ments in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems. UsingtheFourierdevelopmentofsolutionsandtechniquesofNonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total length of the network in a suitable Hilbert space that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree, we characterize these weights in terms of the eigenvalues of the corresponding elliptic problem. The resulting weighted observability inequality allows id- tifying the observable energy in Sobolev terms in some particular cases. That is the case, for instance, when the network is star-shaped and the ratios of the lengths of its strings are algebraic irrational numbers.

• control theory
• multi-structures
• partial differential equation
• partial differential equations on graphs
• string-network
• wave equation
• partial differential equations

From the reviews of the first edition:

"This book deals with propagation, observation and control of the vibrations in a l-d model of a multi-body structure consisting of a finite number of flexible strings distributed along a planar graph. … this book provides a source of information for researchers in the area of control and observation of networks. It also can be used as an introductory textbook for graduate students entering the field." (Claudio Giorgi, Mathematical Reviews, Issue 2006 h)

"The study of mechanical systems consisting of elastic elements is of interest in many situations. … The present book studies the one-dimensional model … . The book is of interest to graduate students and researchers seeking to get an insight into the control theory of elastic systems constituting networks in a rigorous manner. … The book should be quite useful to researchers as a source of recent results and references as well as a self-contained treatment of the subject." (Fiazud Din Zaman, Zentralblatt MATH, Vol. 1083, 2006)

• Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid, Spain

  René Dáger

• Departamento de Matemáticas Facultad de Ciencias, C-XV, Universidad Autónoma de Madrid Cantoblanco, Madrid, Spain

  Enrique Zuazua

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