Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

TL;DR

This paper establishes nonlinear small-gain theorems ensuring input-to-state stability for infinite heterogeneous networks, extending finite network results and providing criteria for linear gain operators.

Contribution

It introduces new nonlinear small-gain conditions for infinite networks, generalizing finite network results and linking spectral radius conditions to stability.

Findings

Network is input-to-state stable under certain small-gain conditions.

Equivalent conditions for finite and infinite networks are established.

Criteria for stability with linear and homogeneous gain operators are provided.

Abstract

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.