The stability of an inverted pendulum when there are rapid random oscillations of the suspension point

The stability of the upper equilibrium position of a pendulum when the suspension point makes rapid random oscillations of small amplitude, is investigated. A class of random oscillations that make the system stable with unit probability for small friction is indicated. It is shown that, if there is no friction, there is no stability, which, as is well known, is not the case for harmonic oscillations of the suspension point. Some general results concerning the impossibility of stochastic stabilization of Hamiltonian systems are proved.