Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping (Q2839312)

The author studies the asymptotic stability of a pendulum with time-varying friction described by the equation NEWLINE\[NEWLINE x^{\prime\prime}+h(t)x^\prime + \sin x = 0, \tag{P} NEWLINE\]NEWLINE where \( h\) is continuous and nonnegative for \( t \geq 0\).NEWLINENEWLINEThe main result is a criterion for the asymptotic stability.NEWLINENEWLINETheorem. Suppose that there exists a \(\gamma_0\) with \(0<\gamma_0<\pi\) such that NEWLINE\[NEWLINE {\liminf}_{t\to\infty}\int_t^{t+\gamma_0}h(s)\,ds>0. NEWLINE\]NEWLINE Then the origin of \((P)\) is asymptotically stable if and only if NEWLINE\[NEWLINE \int_0^{\infty}\frac{\int_0^te^{H(s)}\,ds}{e^{H(t)}}\, dt=\infty, NEWLINE\]NEWLINE where \(H(t)=\int_0^th(s)\,ds\).NEWLINENEWLINEThe proof is based on Lyapunov's stability theory and phase plane analysis of the positive orbits of an planar system equivalent to equation \((P)\).