A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.