On passive stabilization in critical cases
type: Item
instance of: scholarly article
title: On passive stabilization in critical cases
zbMATH Open document ID: 0990.70015
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-08-14T00:00:00Z
review text: It is well known that a (practically) lossless mechanical system can be stabilized by coupling it to a system which contains a friction device (standard practice for terrestrial vehicles). Here the equations of motion of such a system are derived from a Hamiltonian with a damping term. The resulting system of fourth-order ODEs exhibits a Lyapunov critical case with two purely imaginary eigenvalues. Non-relevant terms are removed by a polynomial transformation. The details of this procedure can be found in chapter 4 of \textit{I. G. Malkin}'s book [Theorie der Stabilität der Bewegung. (Russian), Moskau: Verlag `Nauka'. Hauptredaktion für physikalisch-mathematische Literatur (1966; Zbl 0136.08502)] which extends an earlier publication by \textit{Frommer} [Math. Ann. 99, (1928)] which, in turn, is based on the works of I. Bendixson (1900) and H. Poincaré (1881).
Mathematics Subject Classification ID: 70K20
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 1441390
zbMATH Keywords: passive stabilization
zbMATH Keywords: friction device
zbMATH Keywords: equations of motion
zbMATH Keywords: Hamiltonian
zbMATH Keywords: damping term
zbMATH Keywords: Lyapunov critical case
zbMATH Keywords: purely imaginary eigenvalues
zbMATH Keywords: polynomial transformation
reviewed by: Igor Gumowski
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6692
OpenAlex ID: W2052666973
cites work: Q5328302
cites work: Stability theory by Liapunov's direct method
cites work: Sulla ricerca di una funzione di Liapounoff per un sistema differenziale interessante la Meccanica dei sistemi olonomi
cites work: Q4174830
DOI: 10.1006/JMAA.1999.6692
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description: scientific article; zbMATH DE number 1441390