A Lyapunov-based small-gain theorem for infinite networks

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Kawan, Christoph ; Mironchenko, Andrii ; Swikir, Abdalla ; Noroozi, Navid ; Zamani, Majid:
A Lyapunov-based small-gain theorem for infinite networks.
In: IEEE Transactions on Automatic Control. Bd. 66 (2021) Heft 12 . - S. 5830-5844.
ISSN 1558-2523
DOI: https://doi.org/10.1109/TAC.2020.3042410

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Abstract

This article presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially input-to-state stable. The effectiveness of our result is illustrated through several examples including nonlinear spatially invariant systems with sector nonlinearities and a road traffic network.