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Kawan, Christoph; Mironchenko, Andrii; Swikir, Abdalla; Noroozi, Navid and Zamani, Majid (2021): A Lyapunov-Based Small-Gain Theorem for Infinite Networks. In: IEEE Transactions on Automatic Control, Vol. 66, No. 12: pp. 5830-5844

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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.

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