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

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