2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79505This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of ``norm-estimators'', and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.Many related papers can be found in: http://www.math.rutgers.edu/~sontagOptimization and Control93D20;93C15;93B36text