Stochastic robustness of interconnected nonlinear systems in an iISS framework

In this paper, the problem of verifying integral input-to-state stability (iISS) and input-to-state stability (ISS) of interconnected stochastic systems is addressed. The aim is at extending the deterministic iISS small-gain framework to stochastic systems by including ISS as a special case. This paper highlights fundamental differences between the stochastic and deterministic cases. Dealing with nonlinearities in interconnection gives rise to a unique issue in the presence of stochastic noises. The key is to cope effectively with the gradient of Lyapunov functions of subsystems in composing a Lyapunov function of the entire system. It is demonstrated that the influence of the gradient appears considerably different in two types of robustness notions. This paper also clarifies the influence of the gradient on the establishment of a trajectory estimate of iISS which is weaker than ISS.