The Stability of Stochastic Coupled Systems with Time-Varying Coupling and General Topology Structure

We introduce a class of novel stochastic coupled systems in which the coupling structure is time-varying and the topology structure is not strongly connected, and first establish the system on a digraph with a time-varying weight matrix. Motivated by Du and Li (2014), we give a hierarchical method to deal with digraphs without strong connectivity and establish the corresponding hierarchical algorithm to realize this approach. Also, an example is given to illustrate our hierarchical algorithm and its feasibility. In the sequel, based on the theory of asymptotically autonomous systems, Kirchhoff's matrix tree theorem, and Lyapunov method, several moment exponential stability criteria are presented, including a Lyapunov-type theorem and a coefficient-type criterion. Furthermore, theoretical results are applied to stochastic coupled oscillators with time-varying coupling structure (SCTCS), and the stability criterion of SCTCS is obtained. Finally, the effectiveness of theoretical results is illustrated by two numerical examples.