Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping

Abstract

The oscillator $$ x''+h(t)x'+x=0 $$ is considered, where the damping $h:{\Bbb R}_+\to{\Bbb R}_+$ is piecewise continuous and large in the sense $$ \liminf_{t\to\infty}\int _t^{t+\delta}h>0 \quad \text{ for every }\ \delta>0. $$ The problem of intermittent damping, initiated by P\. Pucci and J\. Serrin, is investigated. Let a sequence $\{I_n=[\alpha_n,\beta_n]\}$ of disjoint intervals be given such that $\alpha_n\to\infty$ as $n\to\infty$. A necessary and sufficient condition is given for $\{I_n\}$ and $h$ on $I:=\bigcup _{n=1}^\infty I_n$ guaranteeing $x(t)\to 0$, $x'(t)\to 0$ as $t\to\infty$ for every solution $x$, any way $h$ may be defined out of $I$.