The necessity of the small-gain theorem for time-varying and nonlinear systems

It is shown that an appropriately modified version of the small-gain theorem which uses the `asymptotic gain´ of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of: (1) nonlinear plants subject to nonlinear time-varying perturbations; and (2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient