A small-gain approach to ISS of infinite networks with homogeneous gain operators

We present a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. A key assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator constructed from the internal gains is assumed to be subadditive and homogeneous. This covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. We formulate the small-gain condition in terms of a generalized spectral radius of the gain operator. The effectiveness of our results are illustrated through an example. Particularly, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some kind of symmetry.