On stabilizability of the upper equilibrium of the asymmetrically excited inverted pendulum

摘要

Using purely elementary methods, necessary and sufficient conditions are given for the existence of T-periodic and 2T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating vertically with asymmetric high frequency. The equation of the motion is of the form ¨θ ? 1 l (g + a(t)) θ = 0, where a(t) := ( Ah , if kT ≤ t < kT + Th , ?Ae , if kT + Th ≤ t < (kT + Th ) + Te , (k = 0, 1, . . .); Ah , Ae , Th , Te are positive constants (Th + Te = T); g and l denote the acceleration of gravity and the length of the pendulum, respectively. An extended Oscillation Theorem is given. The exact stability regions for the upper equilibrium are presented.