Non-conservative small-gain conditions for input-to-state stability with respect to not-necessarily compact sets

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Noroozi, Navid ; Geiselhart, Roman ; Grüne, Lars ; Rüffer, Björn ; Wirth, Fabian:
Non-conservative small-gain conditions for input-to-state stability with respect to not-necessarily compact sets.
Passau , 2016 . - 16 S.

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Abstract

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of the 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 not-necessarily compact sets. Toward this end, we first develop a Lyapunov characterization of ωISS via finite-step ωISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee ωISS of a network of systems. Finally, applications of our results to partial input-to-state stability, ISS of time-varying systems, synchronization problems and decentralized observers are given.