On the asymptotic stability for intermittently damped nonlinear oscillators

Abstract

The second order nonlinear differential equation \begin{equation*} x''+h(t,x,x')x'+f(x)=0 \qquad \bigl(x\in\mathbb{R},\ t\in\mathbb{R}_+:=[0,\infty),\ ()':=\tfrac{\text{d}}{\text{d}t}()\bigr), \end{equation*} and a sequence $\{I_n\}_{n=1}^\infty$ of non-overlapping intervals are given, where the damping coefficient $h$ admits an estimate \begin{equation*} a(t)|y|^\alpha w(x,y)\le h(t,x,y)\le b(t) W(x,y)\qquad (t\in I:=\cup_{n=1}^\infty I_n; x,y\in \mathbb{R}). \end{equation*} It is known that if the equation is linear ($f(x)\equiv x$, $h(t,x,x')\equiv h(t)$, $a(t)\le h(t)\le b(t)$), $a(t)\ge \underline{a}=\text{const.}>0$ and $b(t)\le \overline{b}=\text{const.}0$ and/or $\overline{b}<\infty$ do not exist.

Keywords

• intermittent damping
• asymptotic stability
• total mechanical energy
• dissipation
• differential inequalities