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Output-to-State Stability (OSS) is a notion of detectability for nonlinear systems that is formulated in the ISS  framework. We generalize the notion of OSS for systems which possess a decomposable invariant set and evolve on compact manifolds. Building upon a recent extension of the ISS theory for this very class of systems [2], the paper provides equivalent characterizations of the OSS property in terms of asymptotic estimates of the state trajectories and, in particular, in terms of existence of smooth Lyapunov-like functions.

Smooth Output-to-State Stability for multistable systems on compact manifolds