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

Hatvani, L.

doi:10.1090/S0002-9939-96-03266-2

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.

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