A small-gain theory for infinite networks via infinite-dimensional gain operators

TL;DR

This paper introduces a novel small-gain theoretical framework for infinite networks of dynamical systems using infinite-dimensional gain operators, enabling stability analysis through fixed points and decay paths.

Contribution

It develops a new approach to analyze infinite networks via augmented gain operators and fixed points, extending small-gain theory to infinite-dimensional settings.

Findings

Existence of fixed points under a no-joint-increase condition

Construction of decay paths for stability analysis

Recovery of classical results for max-type and subadditive operators

Abstract

In this paper, we develop a new approach to study gain operators built from the interconnection gains of infinite networks of dynamical systems. Our focus is on the construction of paths of strict decay which are used for building Lyapunov functions for the network and thus proving various stability properties, including input-to-state stability. Our approach is based on the study of an augmented gain operator whose fixed points are precisely the points of decay for the original gain operator. We show that plenty of such fixed points exist under a uniform version of the no-joint-increase condition. Using these fixed points to construct a path of strict decay, in general, requires specific dynamical properties of associated monotone operators. For particular types of gain operators such as max-type operators and subadditive operators, these properties follow from uniform global asymptotic stability of the system induced by the original gain operator. This is consistent with former results in the literature which can readily be recovered from our theory.