Non-conservative discrete-time ISS small-gain conditions for closed sets

TL;DR

This paper unifies and generalizes small-gain theory to provide necessary and sufficient conditions for input-to-state stability with respect to closed sets, extending existing results to broader scenarios.

Contribution

It introduces a comprehensive framework for small-gain conditions that ensure ISS with respect to closed sets, including a Lyapunov characterization and applications to various stability problems.

Findings

Provides necessary and sufficient small-gain conditions for $ ext{ISS}$ with respect to closed sets.

Develops a Lyapunov-based characterization of $ ext{ωISS}$.

Applies the theory to partial stability, time-varying systems, and distributed observers.

Abstract

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $\omega$ISS via finite-step $\omega$ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $\omega$ISS of a network of systems. Finally, applications of our results to partial input-to-state stability, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.