The inverted pendulum is a widely used benchmark for testing control algorithms (PID controller, neural networks, Genetic algorithms, etc). Modern controllers are capable of even balancing multiple linked pendulums.
An inverted pendulum may be stabilized by oscillating the support rapidly up and down. If the oscillation is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from pertubations in a strikingly counterintuitive manner.
In practice, the inverted pendulum is frequently made of an aluminium strip, mounted on a ball-bearing pivot; the oscillatory force is conveniently applied with a jigsaw.
If the driving point moves in simple harmonic motion, the pendulum's motion is described by the Mathieu equation.