In this paper we consider the stability of networks consisting of nonlinear ISS systems supplied with ISS Lyapunov functions defined in dissipative form. The problem of constructing an ISS Lyapunov function for the network is addressed. Our aim is to provide a geometrical condition of a small gain type under which this construction is possible and to describe a method of an explicit construction of such an ISS Lyapunov function. In the dissipative form, the geometrical approach allows us to discuss both Lipschitz continuous construction and continuously differentiable construction of ISS Lyapunov functions.

This paper gives a solution to the problem of verifying stability of networks consisting of integral input-to-state stable (iISS) subsystems. The iISS small-gain theorem developed recently has been restricted to interconnections of two subsystems. For large-scale systems, stability criteria relying only on gain-type information that were previously developed address only input-to-state stable (ISS) subsystems. To address the stability problem involving iISS subsystems interconnected in general structure, this paper shows how to construct Lyapunov functions of the network by means of a sum of nonlinearly rescaled individual Lyapunov functions of subsystems under an appropriate small-gain condition.