A cyclic small-gain condition and an equivalent matrix-like criterion for iISS networks

This paper considers nonlinear dynamical networks consisting of individually iISS (integral input-to-state stable) subsystems which are not necessarily ISS (input-to-state stable). Stability criteria for internal and external stability of the networks are developed in view of both necessity and sufficiency. For the sufficiency, we show how we can construct a Lyapunov function of the network explicitly under the assumption that a cyclic small-gain condition is satisfied. The cyclic small-gain condition is shown to be equivalent to a matrix-like condition. The two conditions and their equivalence precisely generalize some central ISS results in the literature. Moreover, the necessity of the matrix-like condition is established. The allowable number of non-ISS subsystems for stability of the network is discussed through several necessity conditions.