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

 

Integral conditions on the asymptotic stability
for the damped linear oscillator
with small damping


Author: L. Hatvani
Journal: Proc. Amer. Math. Soc. 124 (1996), 415-422
MSC (1991): Primary 34D20, 34A30
MathSciNet review: 1317039
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Abstract: The equation $x''+h(t)x'+k^2x=0$ is considered under the assumption $0\le h(t)\le \overline{h}<\infty$ $(t\ge 0)$. It is proved that $\limsup _{t \to \infty}\left(t^{-2/3}\int_0 ^t h\right)>0$ is sufficient for the asymptotic stability of $x=x'=0$, and $2/3$ is best possible here. This will be a consequence of a general result on the intermittent damping, which means that $h$ is controlled only on a sequence of non-overlapping intervals.


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Additional Information

L. Hatvani
Affiliation: Bolyai Institute, Aradi vértanúk tere 1, Szeged, Hungary, H–6720
Email: hatvani@math.u-szeged.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03266-2
PII: S 0002-9939(96)03266-2
Keywords: Intermittent damping, energy method
Received by editor(s): January 7, 1994
Additional Notes: The author was supported by the Hungarian National Foundation for Scientific Research with grant number 1157
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society