New criteria of almost sure exponential stability and instability of nonlinear stochastic systems with a generalization to stochastic coupled systems

This paper focuses on the almost sure exponential stability and instability problems for nonlinear stochastic systems. In contrast with previous works, combining with Lyapunov's second method and inequalities techniques, two classes of less conservative criteria are obtained, depending on piecewise continuous scalar functions (PCSFs). It is highly desirable that the PCSF can not only replace the constant coefficient of the upper bound estimation which belongs to the diffusion operator of a Lyapunov function in available literature but also even be unbounded. Especially, when PCSFs are assumed to be periodic, our theoretical results can be transformed into the case of almost sure exponential stability and instability of nonlinear systems with intermittent noises, and corresponding criteria are derived. Furthermore, the sufficient criteria we present are popularized to stochastic coupled systems taking advantage of graph theory and Lyapunov method. As an application of theoretical results, we put forward an exploration to the synchronization analysis of a class of single-link robot arms system via intermittent noise control. Finally, several numerical examples consisting of a time-varying stochastic system and a stochastic coupled oscillators system as well as single-link robot arms systems are provided to illustrate the feasibility of the stated theoretical results.