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

Abstract

The equation x + h ( t ) x ′ + k 2 x = 0 x+h(t)x’+k^2x=0 is considered under the assumption 0 ≤ h ( t ) ≤ h ¯ > ∞ 0\le h(t)\le \overline {h}>\infty ( t ≥ 0 ) (t\ge 0) . It is proved that lim sup t → ∞ ( t − 2 / 3 ∫ 0 t h ) > 0 \limsup _{t \to \infty }\left (t^{-2/3}\int _0 ^t h\right )>0 is sufficient for the asymptotic stability of x = x ′ = 0 x=x’=0 , and 2 / 3 2/3 is best possible here. This will be a consequence of a general result on the intermittent damping, which means that h h is controlled only on a sequence of non-overlapping intervals.