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.