Finite-time stability theorems of homogeneous stochastic nonlinear systems

The purpose of this paper is to study finite-time stability of a class of homogeneous stochastic nonlinear systems modeled by stochastic differential equations. An existence result of weak solutions for stochastic differential equations with continuous coefficients is derived as a preparation for discussing stochastic nonlinear systems. Then a generalization of finite-time stochastic stability theorem is given. By using some properties of homogeneous functions and homogeneous vector fields, it is proved that a homogeneous stochastic nonlinear system is finite-time stable if its coefficients have negative degrees of homogeneity, and there exists a sufficiently smooth and homogeneous Lyapunov function such that the infinitesimal generator of the stochastic system acting on it is negative definite. In the case when the drift coefficient of a stochastic system is homogeneous and has a negative degree of homogeneity, it can be shown that the stochastic system is also finite-time stable under appropriate conditions. Two examples are provided as illustrations.