A Lyapunov-based ISS small-gain theorem for infinite networks of nonlinear systems

TL;DR

This paper establishes conditions under which an infinite network of interconnected ISS nonlinear systems remains stable, by extending Lyapunov and small-gain theorems to infinite-dimensional settings.

Contribution

It introduces a Lyapunov-based ISS small-gain theorem for infinite networks, linking stability to properties of an infinite-dimensional nonlinear operator.

Findings

Infinite ISS networks admit Lyapunov functions under weak coupling.

UGAS of the associated operator ensures network stability.

Necessary and sufficient conditions relate to small-gain criteria.

Abstract

In this paper, 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.