On Input-to-State Stability with respect to decomposable invariant sets

We propose a global notion of Input-to-State Stability (for nonlinear systems evolving on manifolds) in the form of an asymptotic gain condition with respect to the Riemannian distance to a compact invariant set. The invariant set is assumed to admit a decomposition without cycles (basically no homoclinic nor heteroclinic orbits may exist). The notion is flexible enough to allow for unstable sets and yet is suitable for a Lyapunov-like characterization that will be discussed. Applications can be envisaged also in the context of the analysis of incremental stability on manifolds.