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

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- by L. Hatvani
- Proc. Amer. Math. Soc.
**124**(1996), 415-422 - DOI: https://doi.org/10.1090/S0002-9939-96-03266-2
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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.## References

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## Bibliographic Information

**L. Hatvani**- Affiliation: Bolyai Institute, Aradi vértanúk tere 1, Szeged, Hungary, H–6720
- MR Author ID: 82460
- Email: hatvani@math.u-szeged.hu
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**124**(1996), 415-422 - MSC (1991): Primary 34D20, 34A30
- DOI: https://doi.org/10.1090/S0002-9939-96-03266-2
- MathSciNet review: 1317039