SMITH-TYPE STABILITY THEOREMS FOR THE DAMPED LINEAR OSCILLATOR

Sufficient conditions are given guaranteeing that every solution of the equation

\[x''+h(t)x'+\omega^2x=0 \qquad (h(t)\ge 0,\ x\in {\mathbb{R}})\]

and its derivative tend to zero as $t\to\infty$. The results are applicable in the general case $0\le 0\le h(t)<\infty$, i.e., conditions $h(t)\ge$const.$>0$ and $h(t)\le$const.$<\infty$ are not required in general. In the first main theorem the damping is controlled on the whole half-line $[0,\infty)$. The second main theorem is devoted to the problem of the intermittent damping, when  conditions are supposed only on the union of non-overlapping intervals.