## Differential and Integral Equations

### 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 $I$.

#### Article information

Source
Differential Integral Equations, Volume 10, Number 2 (1997), 265-272.

Dates
First available in Project Euclid: 2 May 2013

https://projecteuclid.org/euclid.die/1367526337

Mathematical Reviews number (MathSciNet)
MR1424811

Zentralblatt MATH identifier
0893.34045

Subjects
Primary: 93D15: Stabilization of systems by feedback

#### Citation

Hatvani, László; Krisztin, Tibor. Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping. Differential Integral Equations 10 (1997), no. 2, 265--272. https://projecteuclid.org/euclid.die/1367526337