On a Small Gain Theorem for ISS Networks in Dissipative Lyapunov Form

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.