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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping


Author: Jitsuro Sugie
Journal: Proc. Amer. Math. Soc. 141 (2013), 2419-2427
MSC (2010): Primary 34D23, 34D45; Secondary 34C15, 37C70
Published electronically: March 28, 2013
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Abstract: A necessary and sufficient condition is given for the asymptotic stability of the origin of a pendulum with time-varying friction described by the equation

$\displaystyle x'' + h(t)x' + \sin x = 0, $

where $ h(t)$ is continuous and nonnegative for $ t \ge 0$. This condition is expressed as a double integral on the friction $ h(t)$. The method that is used to obtain the result is Lyapunov's stability theory and phase plane analysis of the positive orbits of an equivalent planar system to the above-mentioned equation.

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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
Email: jsugie@riko.shimane-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11615-1
PII: S 0002-9939(2013)11615-1
Keywords: Asymptotic stability, phase plane analysis, nonlinear oscillator, damped pendulum, growth condition
Received by editor(s): October 20, 2011
Published electronically: March 28, 2013
Additional Notes: This work was supported in part by Grant-in-Aid for Scientific Research No.22540190 from the Japan Society for the Promotion of Science
Communicated by: Yingfei Yi
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.