  We propose a time-discounted integral variant of incremental
input/output-to-state stability (i-iIOSS) together with an equivalent Lyapunov
function characterization. Continuity of the i-iIOSS Lyapunov function is
ensured if the system satisfies a certain continuity assumption involving the
Osgood condition. We show that the proposed i-iIOSS notion is a necessary
condition for the existence of a robustly globally asymptotically stable
observer mapping in a time-discounted ``$L^2$-to-$L^\infty$'' sense. In
combination, our results provide a general framework for a Lyapunov-based
robust stability analysis of observers for continuous-time systems, which in
particular is crucial for the use of optimization-based state estimators (such
as moving horizon estimation).
