Stability of a mechanical system with unbounded dissipative forces
In this article we shall be concerned with a mechanical system described by the Lagrangian equation \[ (1)\quad(d/dt)(\partial T/\partial \dot q)- \partial T/\partial q=-\partial \Pi /\partial q-B(t,q)\dot q+G(t,q)\dot q \] with generalized coordinates \(q\in R^ n\) and generalized velocities \(\dot q\in R^ n\), and show that the equilibrium \(q=\dot q=0\) of (1) is weakly uniformly asmyptotically stable under some familiar conditions and the following; for any bounded continuous function \(\psi\) (s) on \([0,\infty)\) there exist a sequence of positive numbers \(\{s_ n\}\) and a positive constant d, \(s_{n+1}\geq s_ n+d\), such that tr B(s,\(\psi\) (s))\(\not\equiv 0\) on \([s_ n,s_ n+d]\) for all n and that \[ \sum^{\infty}_{n=1}[\int^{s_ n+d}_{s_ n}tr B(s,\psi(s))ds]^{-1}=\infty, \] where tr B(s,\(\psi\) (s)) denotes the trace of the matrix B(s,\(\psi\) (s)). It should be noted that \(B(t,q)=tE\) (E is the unit matrix in \(R^{n\times n})\) and G(t,q)\(\equiv 0\) do not satisfy the condition considered by \textit{L. Hatvani} [Colloq. Math. Soc. János Bolyai 30, 381-409 (1981; Zbl 0486.34044)]. However, our result is applicable to a mechanical system with unbounded B satisfying \(0<tr B\leq Mt\cdot \log(1+t)\), \(t\geq 0\), for some positive constants M and N.
weakly uniformly asymptotically stable equilibrium
scientific article; zbMATH DE number 3847830