A Lyapunov-based small-gain theorem for infinite networks

TL;DR

This paper introduces a Lyapunov-based small-gain theorem for infinite networks, establishing conditions for exponential input-to-state stability based on spectral radius criteria, with applications to nonlinear systems and traffic networks.

Contribution

It provides a novel small-gain theorem for infinite networks using Lyapunov functions and spectral radius conditions, extending stability analysis to countably infinite systems.

Findings

The gain operator’s spectral radius being less than one guarantees stability.

The theorem applies to nonlinear spatially invariant systems with sector nonlinearities.

Examples include road traffic networks demonstrating the theorem's practical relevance.

Abstract

This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially input-to-state stable. The effectiveness of our result is illustrated through several examples including nonlinear spatially invariant systems with sector nonlinearities and a road traffic network.