Stability analysis of dynamical systems randomized by multi-dimensional Wiener processes

In this paper, we show that stochastic asymptotic stability supplied by “stabilization by noise” is an extended version of deterministic asymptotic stability. To solve the problem, we first revisit the randomization problem of nonlinear dynamical systems by adding multi-dimensional Wiener processes. We also summarize uniform almost sure asymptotic stability (UASAS) is almost the same as asymptotic stability for deterministic systems. Then, we clarify necessary and sufficient conditions for the origins of randomized systems to be UASAS. Furthermore, we also discuss the possibility of the stabilization by noise with considering the randomization problem and UASAS property.