Koopman Operators for Global Analysis of Hybrid Limit-Cycling Systems: Construction and Spectral Properties

This paper reports a theory of Koopman operators for a class of hybrid dynamical systems with globally asymptotically stable periodic orbits, called hybrid limit-cycling systems. We leverage smooth structures intrinsic to the hybrid dynamical systems, thereby extending the existing theory of Koopman operators for smooth dynamical systems. Rigorous construction of an observable space is carried out to preserve the inherited smooth structures of the hybrid dynamical systems. Complete spectral characterization of the Koopman operators acting on the constructed space is then derived where the existence and uniqueness of their eigenfunctions are ensured. Our results facilitate global analysis of hybrid dynamical systems using the Koopman operator.