On a small gain theorem for networks of iISS systems

This paper considers networks consisting of integral input-to-state stable (iISS) subsystems and addresses the problem of verifying iISS property of a given network. First, we focus on construction of continuously differentiable Lyapunov functions, and derive a condition ensuring the iISS of the network comprising n subsystems. Although this approach referred to as the sum-type construction has not yet been reduced to an easily computable condition for general n, the n = 2 case recovers the iISS small-gain condition for two subsystems developed recently. Next, in the case of n subsystems, using Lipschitz continuous Lyapunov functions, this paper derives a small-gain condition. It is shown that this second approach referred to as the max-type construction fails to offer a Lyapunov function if there exist subsystems which are not input-to-state stable (ISS). The relation between the two formulations is discussed in the case of two ISS subsystems.