ISS small-gain criteria for infinite networks with linear gain functions

This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed infinitely many finite-dimensional systems. We model these on infinite-dimensional ? ? -type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling influence subsystems each other, are linear functions. Moreover, gain operator built from gains assumed to be subadditive and homogeneous, which covers both max-type sum-type formulations ISS functions subsystems. As consequence, condition can formulated terms generalized spectral radius operator. By an example, we show easily checked if interconnection topology network has some sort symmetry. While main result function implication form overall network, dissipative constructed under mild extra assumptions.