  We consider a possible application of the Wa\.zewski topological method to
feedback control systems and to more general dynamical systems. We show how
this method can be used to prove the impossibility of global stabilization in
such problems. Moreover, we give sufficient conditions for the existence of a
solution such that its trajectory never leaves a subset of the extended phase
space of the system and does not tend asymptotically to a given equilibrium. We
illustrate our result with various real-life systems including the Furuta
pendulum and the wheeled inverted pendulum.
