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 ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.
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