Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping

Sugie, Jitsuro

doi:10.1090/S0002-9939-2013-11615-1

Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping
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Proc. Amer. Math. Soc. 141 (2013), 2419-2427 Request permission

Abstract:

A necessary and sufficient condition is given for the asymptotic stability of the origin of a pendulum with time-varying friction described by the equation \[ x'' + h(t)x’ + \sin x = 0, \] where $h(t)$ is continuous and nonnegative for $t \ge 0$. This condition is expressed as a double integral on the friction $h(t)$. The method that is used to obtain the result is Lyapunov’s stability theory and phase plane analysis of the positive orbits of an equivalent planar system to the above-mentioned equation.

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