Analysis of different Lyapunov function constructions for interconnected hybrid systems

This paper studies stability of interconnections of hybrid dynamical systems, in the general scenario that the continuous or discrete dynamics of subsystems may have destabilizing effects. We analyze two existing methods of constructing Lyapunov functions for the interconnection based on candidate ISS Lyapunov functions for subsystems, small-gain conditions, and auxiliary timers modeling restrictions on jump frequency in terms of average dwell-time and reverse average dwell-time. We compare their feasibility and limitations for different types of subsystem dynamics, and examine a case that the combination of them is needed to establish global asymptotic stability.