Some problems on the inverted pendulum dynamics

Let us consider the motion of a planar (inverted) pendulum in the presence of a horizontal force, which is not assumed to be bounded (its magnitude can be arbitrarily large as time tends to infinity). Is it true that, for any such force, there exists an initial position of the pendulum such that, being released with zero initial velocity from this position, the pendulum never falls? If this external force is periodic, is it true that there exists a periodic non-falling solution? Do similar statements hold for a spherical inverted pendulum? We will answer these and a number of other questions in our talk.