Necessary conditions for global asymptotic stability of networks of iISS systems

In this paper, necessary conditions are investigated for global asymptotic stability of dynamical networks consisting of integral input-to-state stable (iISS) subsystems. Employing the dissipation formulation for subsystems, this paper naturally covers input-to-state stable (ISS) subsystems, which constitute a strict subset of the iISS. The number n of subsystems composing a network is allowed to be arbitrary. This paper gives answers to the question of how many non-ISS subsystems are allowed in the network, and the question of the necessity of small-gain-type properties. All the developments in this paper are natural, but non-trivial extensions of the results obtained previously for n = 2. This paper also proposes a concise representation of the iISS small-gain criteria for an arbitrary number of subsystems forming a cycle in networks.