Stabilization of Linear Systems by Noise

It is proved that the biggest Lyapunov number λ ₘₐₓ of the system x = (A + F(t))x, where A is a fixed d × d matrix and $F(t)$ is a zero mean strictly stationary matrix-valued stochastic process, satisfies 1 / d trace A λ ₘₐₓ. On the other hand, for each ε > 0 there is a process $F(t)$ for which λ ₘₐₓ 1 / d trace A + ε. In particular, the system x = Ax can be stabilized by zero mean stationary parameter noise if and only if trace $A < 0$. The stabilization can be accomplished by a one-dimensional noise source. The results carry over to the case where A is a stationary process. They are also true for $F(t) = $ white noise.