Necessary conditions for stability of networks of iISS systems

Abstract

This paper investigates networks of integral input-to-state stable (iISS) systems from the perspective of necessary conditions for the overall stability. This paper also covers input-to-state stable (ISS) systems which constitute a subset of the iISS. The number n of subsystems composing a network is allowed to be arbitrary. Investigating cycles existing in the network, this paper derives necessary conditions for global asymptotic stability of an equilibrium of the overall network. The results for cycle graphs are utilized for dealing with general graphs of interconnection. All the developments in this paper are natural but non-trivial extensions of necessary conditions obtained previously for n = 2, which includes the necessity of iISS small-gain properties and ISS components. This paper also presents a compact representation of the iISS small-gain criteria applicable uniformly to the general number of subsystems comprising a cyclic network.