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A two-phase approach to stability of networks given in iISS framework: Utilization of a matrix-like criterion

This article is concerned with global asymptotic stability (GAS) of dynamical networks. The case when subsystems are integral input-to-state stable (iISS) has been recognized as notoriously difficult to deal with in the literature. In fact, the lack of energy dissipation for large input denies direct application of the small-gain argument for input-to-state stable (ISS) subsystems. Here for networks consisting of iISS subsystems it is demonstrated that a two-phase approach allows us to make use of the ISS small-gain argument by separating a trajectory into a transient and a subsequent ISS-like phase. In contrast to the previous iISS results, the two-phase approach immediately leads to a sufficiency criterion for GAS of general nonlinear networks, which is given in a matrix-like form (order condition).