Stabilization of Linear Systems by Noise

It is proved that the biggest Lyapunov number $\lambda _{\max } $ of the system $\dot x = (A + F(t))x$, where $A$ is a fixed $d \times d$ matrix and $F(t)$ is a zero mean strictly stationary matrix-valued stochastic process, satisfies ${1 / d}$ trace $A \leqq \lambda _{\max } $. On the other hand, for each $\varepsilon > 0$ there is a process $F(t)$ for which $\lambda _{\max } \leqq {1 / d}$ trace $A + \varepsilon $. In particular, the system $\dot x = Ax$ 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.