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Stabilization of companion form systems by mean zero noise

It is shown that the Lyapunov exponents of a linear system with companion form d x d-matrix can be squeezed into an arbitrarily small neighborhood of trace A/d by means of mean zero degenerate real noise or a combination of real and white noise. In particular, the damped inverse pendulum can be stabilized by such a noise. The proof includes a random transformation of the coordinate system and a stochastic averaging principle.