Stochastic Stability via Lyapunov Functions without Differentiability at Supposed Equilibria*

The objective of this paper is to tackle point stability of stochastic systems without requiring noise diffusion coefficients to vanish at the point of interest. Behavioral differences between vanishing coefficients and non-vanishing coefficients arising in adding stochastic noises are illustrated by examples. To identify the differences appropriately, this paper examines the concept of equilibria and proposes several stability properties by introducing the notions of instantaneous points and almost sure equilibria. In addition to clarifying the relationship between those stability properties, Lyapunov-type characterizations are presented as sufficient conditions for those stability properties by making use of functions which are not necessarily twice differentiable at the point of interest. Discussion is also given to relate the stability property of an instantaneous point with noise-to-state stability.