Integral input-to-state stability of interconnected iISS systems by means of a lower-dimensional comparison system

We consider arbitrarily many interconnected integral Input-to-State Stable (iISS) systems in an arbitrary interconnection topology and provide an (i)ISS comparison principle for networks. We show that global asymptotic stability of the origin (GAS) of a lower-dimensional system termed the comparison system, which is based on the individual dissipative Lyapunov iISS inequalities, together with a scaling condition implies the existence of an iISS Lyapunov function of the composite system. A sufficient (but not necessary) condition for 0-GAS of the interconnection is shown in this paper to be the generalized small-gain condition derived by Dashkovskiy et al., but this time in a dissipative Lyapunov setting. We also provide geometric intuition behind growth rate conditions for the stability of cascaded iISS systems.