Applications of the general Lyapunov ISS small-gain theorem for networks

In the framework of the ISS Lyapunov formulation a small gain theorem has recently been proved which allows the explicit construction of Lyapunov functions for interconnected systems. In this note we recall the definitions of ISS Lyapunov functions and the corresponding general small gain theorems. These are then exemplarily used to prove input-to-state stability of and to construct ISS Lyapunov functions for four areas of applications: linear systems, a Cohen-Grossberg neuronal network, error dynamics in formation control, as well as nonlinear transistor-linear resistor circuits.