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Motivated by a paradigm shift towards a hyper-connected world, we develop acomputationally tractable small-gain theorem for a network of infinitely manysystems, termed as infinite networks. The proposed small-gain theorem addressesexponential input-to-state stability with respect to closed sets, which enablesus to analyze diverse stability problems in a unified manner. The small-gaincondition, expressed in terms of the spectral radius of a gain operatorcollecting all the information about the internal Lyapunov gains, can benumerically computed for a large class of systems in an efficient way. Todemonstrate broad applicability of our small-gain theorem, we apply it to thestability analysis of infinite time-varying networks, to consensus ininfinite-agent systems, as well as to the design of distributed observers forinfinite networks.

Small-gain theorem for stability, cooperative control and distributed observation of infinite networks