On vector Lyapunov functions for stochastic dynamical systems

Vector Lyapunov functions are used in the stability analysis of large-scale stochastic systems described by Itô differential equations (with stochastic disturbances in the subsystems and in the interconnecting structure). Sufficient conditions for asymptotic stability and exponential stability with probability 1 and in probability are established, in all cases the objective is the same: to analyze large-scale systems in terms of their lower order (and simpler) subsystems and in terms of their interconnecting structure. Use of the method presented makes it often possible to circumvent difficulties usually encountered when the Lyapunov method is applied to high-dimensional systems and to systems with complicated interconnecting structure. In order to demonstrate the usefulness of the present approach, a specific example is considered.