Lyapunov function-based stochastic small-gain theorem and its applications

In this paper, an alternative statement and proof of the so-called stochastic nonlinear small-gain theorem is given by means of Lyapunov function argument. For this end, the infinitesimal generator and the generalized Itpsilas formula for non-smooth Lyapunov function are represented; the existence and uniqueness of strong solution to system are proved by the aid of Lyapunov function whose mathematical expectation is bounded; global asymptotic stability in probability is obtained for stochastic systems by using a non-smooth Lyapunov function. Then the Lyapunov function-based small-gain theorem is proposed by adding other assumptions to the usual small-gain condition, and is proved by adopting ldquoswitching Lyapunov function methodrdquo. As application, we address the problem of adaptive output-feedback stabilization for a class of interconnected nonlinear stochastic systems.