On necessary conditions for stability of interconnected iISS systems

This paper investigates necessary conditions for stability of nonlinear systems made of feedback interconnection of two iISS subsystems. The integral input-to-state stability(iISS) is a dissipative property which includes the input-to-state stability(ISS) as a special case. It is proved that at least one subsystem needs to be ISS for global asymptotic stability of the feedback loop when two subsystems are not completely specified and only their supply rates are available. In contrast, the situation where ODEs describing subsystems are available does not exclude pairs of subsystems which are only iISS. This paper further investigates necessary conditions for global asymptotic stability, and proves that the nonlinear small-gain condition is necessary in the case of unspecified iISS subsystems described with supply rates. When one of the two subsystems is given as a specific ODE, the nonlinear small-gain condition is no longer necessary, while the linear small-gain condition is necessary in the case of linear systems. For nonlinear systems, the necessity recovers only if the supply rate fits the known subsystem tightly in the shape as well as the magnitude.