Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions

This paper proposes a notion of viscosity weak supersolutions to build a bridge between stochastic Lyapunov stability theory and viscosity solution theory. Different from ordinary differential equations, stochastic differential equations can have the origins being stable despite having no smooth stochastic Lyapunov functions (SLFs). The feature naturally requires that the related Lyapunov equations are illustrated via viscosity solution theory, which deals with non-smooth solutions to partial differential equations. This paper claims that stochastic Lyapunov stability theory needs a weak extension of viscosity supersolutions, and the proposed viscosity weak supersolutions describe non-smooth SLFs ensuring a large class of the origins being noisily (asymptotically) stable and (asymptotically) stable in probability. The contribution of the non-smooth SLFs are confirmed by a few examples; especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled systems have the origins being noisily asymptotically stable for any additive noises.