Non-uniform ISS small-gain theorem for infinite networks

TL;DR

This paper introduces a non-uniform input-to-state stability concept for infinite networks, showing that under a uniform small-gain condition, the entire network and its finite subnetworks are stable, even without uniform transient bounds.

Contribution

It extends the small-gain theorem to infinite networks with non-uniform convergence, providing new stability conditions for complex interconnected systems.

Findings

Infinite networks with non-uniform stability are stable under the small-gain condition.

Finite subnetworks inherit input-to-state stability.

The approach accommodates subsystems without uniform transient bounds.

Abstract

We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork, while it allows for non-uniform convergence of all components. For an infinite network consisting of input-to-state stable subsystems, that do not necessarily have a uniform KL-bound on the transient behaviour, we show: If the gain operator satisfies the uniform small-gain condition, then the whole network is non-uniformly input-to-state stable and all its finite subnetworks are input-to-state stable.