Explicit solutions to state-dependent scaling problems for interconnected iISS and ISS nonlinear systems

This paper addresses the problem of establishing global stability properties of nonlinear interconnected systems. The state-dependent scaling problem is presented as a unified mathematical formulation whose solutions explicitly provide Lyapunov functions proving stability properties of feedback and cascade systems. The framework covers diverse nonlinearities represented by general supply rates. In order to let the state-dependent approach have real usefulness beyond formal applicability, this paper derives explicit formulas for solutions to state-dependent scaling problems for integral input-to-state stable (iISS) systems and input-to-state stable (ISS) systems.