Solution concepts and analysis of spatially invariant hybrid systems: Exploring Zeno and beyond

In this paper we consider hybrid systems, consisting of an infinite number of interconnected spatially invariant (identical) hybrid subsystems described using the hybrid inclusions framework. It can be shown that such interconnections can be very useful in, e.g., the modeling of interconnected networked subsystems that use packet-based communication for the exchange of information, including autonomously driving platoons of vehicles. Interestingly however, due to the infinite-dimensionality of the overall interconnected hybrid system, establishing proper definitions of solutions becomes a difficult task as standard solution concepts do not apply to the systems under study since Zeno behavior (an infinite number of jumps in a bounded time interval) is inevitable. Therefore, we introduce an alternative and natural solution concept for this class of systems, allowing solutions to be defined beyond Zeno points. In addition, based on this novel solution concept, we derive Lyapunov-based conditions for a specific, but relevant class of infinite-dimensional hybrid systems, as used for modeling, for instance, networked control systems, that guarantee UGAS (or UGES) or Lp-stability of the overall interconnected system.