Nonlinear Small-Gain Condition Covering iISS Systems: Necessity and Sufficiency from a Lyapunov Perspective

This paper presents conditions for stability of interconnected nonlinear systems consisting of integral input-to-state stable subsystems. Both the necessity and the sufficiency are investigated using a Lyapunov formulation. The ultimate result is a necessary and sufficient condition for integral input-to-state stability (iISS) of the interconnection, which is believed to be the first of its kind. This paper derives two sufficient conditions. One is the iISS-ISS small-gain condition which has been proposed earlier by one of the authors. Its sufficiency is strengthened to cover situations more general than the earlier result. The other condition is a universal representation of the nonlinear small-gain condition which has been a popular tool for dealing with input-to-state stable (ISS) subsystems. This paper proves that, for establishing iISS property of the interconnection, the universal representation can deal with iISS subsystems as much as the iISS-ISS small-gain condition if supply rate functions are smooth everywhere and analytic at zero. Furthermore, two necessary conditions this paper derives demonstrate that the nonlinear small-gain condition in the universal form is also necessary for the iISS property