Construction of Lyapunov Functions for Second-Order Linear Stochastic Stationary Systems

In this paper, we present a construction of Lyapunov functions for second-order linear stochastic systems with constant coefficients. Based on this construction, we state necessary and sufficient conditions of the mean-square exponential stability of two-dimensional linear stationary systems. We obtain analytical expressions for the bifurcation value of the intensity of white noise acting on the parameters of the system. As an example, we consider equations of elastic vibrations whose coefficients are perturbed by white noise.