Uniform asymptotic stability of time-varying damped harmonic oscillators

Ishihara, Kazuki; Sugie, Jitsuro

doi:10.1090/bproc/30

Uniform asymptotic stability of time-varying damped harmonic oscillators
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by Kazuki Ishihara and Jitsuro Sugie HTML | PDF

Proc. Amer. Math. Soc. Ser. B 4 (2017), 31-46

Abstract:

This paper presents sufficient conditions which guarantee that the equilibrium of the damped harmonic oscillator \begin{equation*} x” + h(t)\!\:x’ + \omega ^2x = 0 \end{equation*} is uniformly asymptotically stable, where $h\!: [0,\infty ) \to [0,\infty )$ is locally integrable. These conditions work to suppress the rapid growth of the frictional force expressed by the integral amount of the damping coefficient $h$. The obtained sufficient conditions are compared with known conditions for uniform asymptotic stability. Two diagrams are included to facilitate understanding of the conditions. By giving a concrete example, remaining problems are pointed out.

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