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 function, provided the couplings between subsystems are sufficiently weak. The strength is described in terms properties infinite-dimensional nonlinear positive operator, built from interconnection gains. If operator induces a uniformly globally asymptotically (UGAS) system, function for can be constructed. We analyze necessary and sufficient conditions UGAS relate them to small-gain used stability analysis finite networks.