A Measure-Theoretic Formulation of Hybrid Systems Beyond Zeno Time

Hybrid dynamical systems may exhibit Zeno behavior, where infinitely many discrete transitions occur in finite time, leading to a loss of well-posedness of trajectories beyond the accumulation point. This paper develops a measure-theoretic formulation of hybrid dynamics by representing discrete transitions through finite vector-valued Radon measures and recasting the system as a measure differential inclusion. Within this framework, we establish a closure result for extended hybrid solutions in the space of functions of bounded variation and derive an existence result under suitable approximation assumptions. We also prove consistency with classical hybrid trajectories in the absence of Zeno behavior and characterize the state at the Zeno time as the left limit of the hybrid evolution, together with any additional vector atom deliberately assigned at the accumulation time. The proposed formulation provides a natural basis for continuation beyond the accumulation point and allows Lyapunov-based stability properties to be formulated directly at the level of the measure-driven dynamics. A central feature of the approach is that infinitely many discrete transitions with finite total variation of the jump increments are encoded by a finite vector-valued atomic measure whose atoms may accumulate at the Zeno time. An additional vector atom at the Zeno time may be introduced as a lumped effective jump; however, such an atom should be understood as a modeling choice, not as an automatic consequence of the jump sequence. The results are illustrated on an event-triggered control system exhibiting Zeno accumulation.