A Necessary and Sufficient Condition for Exponential Stability of Large-Scale Stochastic Delay Systems in Hierarchical Form

In this paper we investigate the exponential stability of a large-scale stochastic delay system in hierarchical form, i.e., the system consists of several subsystems and each subsystem interacts only with “lower” subsystems but not with “higher” subsystems. It is shown under a condition, which is weaker than Lipschtiz continuity, that the large-scale system is exponential stable in L² if and only if each of the corresponding isolated subsystems is exponential stable in L². The stability of stochastic systems with constants delay is discussed in more details.