  This 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.
