Stabilization of Linear Systems by a Multiplicative Random Noise

Abstract

We consider a linear system of differential equations multiplicatively perturbed by a random noise and obtain formulas for the expectation and the correlation function of the solution. We show that, under a multiplicative perturbation, a Lyapunov unstable linear system can become asymptotically stable in mean, and a stable linear system can become unstable in mean. The corresponding examples of such systems are given.