Stability under events for a class of hybrid dynamical systems with continuous and discrete time variables

Review of Some Control Theory Results on Uniform Stability of Impulsive Systems

“…The exponential event-ISS is an extension of quasi-exponential ISS. Other related stability results on event-stability including event-ISS can be found in [22,43,[55][56][57].…”

“…In [22], the notion of stability under events (event-stability) was proposed and a basic necessary and sufficient condition was derived for event-GUAS via the method of hybrid-event-time Lyapunov function. Recently, in [55][56][57], more sufficient conditions for event-stability were reported by extending the average dwell time (ADT) [36] to the hybrid ADT. Moreover, by employing the approaches including small-gain method [44][45][46][47], vector Lyapunov functions, Razumikhin-type technique, Lyapunov-Krasovskii functional, and Halanay-type inequalities [48,49], the stability results including ISS for hybrid networks with or without time-delays have also been established, see, e.g., [50][51][52][53][54].…”

“…In the following, due to the limited space, we only focus on some uniform stability results in [60][61][62][63] and the event-stability in [22,57] for impulsive hybrid systems.…”

“…The stability property for the system in Equation (31) belongs to the type of "destabilizing flow + destabilizing jump". This type of stability was called as event-stability in [22,55,57]. The method of hybrid-event-time Lyapunov function proposed in [22] provided a tool in analyzing event-stability including event-ISS for hybrid systems.…”

“…Based on the hybrid-event-time, the notion of stability under events (event-stability) was proposed and a basic result on necessary and sufficient condition for global uniform asymptotic stability under events (event-GUAS) was derived in [22] for discrete-time switched hybrid systems. In [55][56][57], more sufficient conditions for event-stability were reported by extending the average dwell time (ADT) [36] to the hybrid ADT. Note that, in the method via the hybrid ADT condition, the requirement of stable flow or stable jump as in the method via ADT [36] can be relaxed.…”