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

1. Auflage 2005
ISBN: 978-3-540-27239-7
Verlag: Springer

Buch, Englisch, Band 50, 230 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 760 g

Reihe: Mathématiques et Applications

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.

Zielgruppe

Research

Autoren/Hrsg.

Dáger, René

Zuazua, Enrique

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Naturwissenschaften Physik Mechanik Klassische Mechanik, Newtonsche Mechanik

Inhaltsverzeichnis

Preliminaries.- Some Useful Tools.- The Three String Network.- General Trees.- Some Observability and Controllability Results for General Networks.- Simultaneous Observation and Control from an Interior Region.- Other Equations on Networks.- Final Remarks and Open Problems.