  We consider a classical problem of control of an inverted pendulum by means
of a horizontal motion of its pivot point. We suppose that the control law can
be non-autonomous and non-periodic w.r.t. the position of the pendulum. It is
shown that global stabilization of the vertical upward position of the pendulum
cannot be obtained for any Lipschitz control law, provided some natural
assumptions. Moreover, we show that there always exists a solution separated
from the vertical position and along which the pendulum never becomes
horizontal. Hence, we also prove that global stabilization cannot be obtained
in the system where the pendulum can impact the horizontal plane (for any
mechanical model of impact). Similar results are presented for several
analogous systems: a pendulum on a cart, a spherical pendulum, and a pendulum
with an additional torque control.
