Input-to-state-KL-stability and criteria for a class of hybrid dynamical systems

This paper aims to study input-to-state-KL-stability (IS-KL-stability) for a class of hybrid dynamical systems (HDS). The notions of IS-KLK0-property and IS-KL-stability are proposed. The IS-KLK0-property, including input-to-state stability (ISS), IS-KL-stability, and some non-ISS property, is a more general input-to-state property and reflects the specific effects on the dynamic behaviors from the dynamics of subsystems, switching events, and external inputs in HDS. An HDS with IS-KLK0-property is allowed to have non-ISS property. The criteria of IS-KL-stability are established for HDS with IS-KLK0-property subsystems via introducing a hybrid dwell time condition. The hybrid dwell time condition unifies the often used average dwell-time condition and converse average dwell-time condition. Moreover, the criteria on IS-KL-stability including event-ISS are established by the method of multiple Lyapunov-like functions. Finally, two examples are provided to verify the theoretical results.