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

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 \[ x + h ( t ) x ′ + sin ⁡ x = 0 , x + h(t)x’ + \sin x = 0, \] where h ( t ) h(t) is continuous and nonnegative for t ≥ 0 t \ge 0 . This condition is expressed as a double integral on the friction h ( t ) 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.