Set stability of infinite networks: ISS small-gain theory and its applications

Abstract Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for networks of infinitely many subsystems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically checked efficiently for a large class of systems. To demonstrate broad applicability of our small-gain theorem, we apply it to consensus of infinitely many agents, and to the design of distributed observers for infinite networks.