Input-to-State Stability for cascade systems with decomposable invariant sets

In a recent paper, the notion of Input-to-State Stability (ISS) has been generalized for systems with decomposable invariant sets and evolving on Riemannian manifolds. In this work, we analyze the cascade interconnection of such ISS systems, we characterize the finest possible decomposition of its invariant set, and we provide the admissible gain for its ISS stability. Specifically, the following two scenarios are considered: 1. the driving system exhibits multistable behavior (fixed points only); 2. the driving system oscillates or rests (periodic orbits and fixed points) while assuming the incremental ISS of the driven system. Furthermore, marginal results on the backward/forward asymptotic behavior of incremental ISS systems are presented. A simple example illustrates the potentiality of the proposed framework.