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

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.