A small-gain theorem for set stability of infinite networks: Distributed observation and ISS for time-varying networks

We generalize a small-gain theorem for a network of infinitely many systems, recently developed in [Kawan et. al, IEEE TAC (2021)]. The generalized small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to study diverse control-theoretic problems in a unified manner, and it also allows for agents to have infinitely many neighbors. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, has several useful characterizations which simplify the verification of the condition. To demonstrate applicability of our small-gain theorem, we apply it to the stability analysis of infinite time-varying networks as well as the design of distributed observers for infinite networks.