Sum-separable Lyapunov functions for networks of ISS systems: A gain function approach

It has been known for a decade now that for networks of input-to-state stable (ISS) systems a Lyapunov function for the network can be constructed as a maximum of re-scaled Lyapunov functions of the subsystems. In this work it is shown that -under the same conditions- instead one could as well choose a sum of re-scaled Lyapunov functions, which is not inherently non-smooth, thus answering a long open question. Moreover, the present approach is constructive, and explicit formulas for the cases of two and three subsystems are stated.