Exponential input-to-state stability under events for hybrid dynamical networks with coupling time-delays

This paper studies the exponential input-to-state stability under events (called as exponential ISS under events) for hybrid dynamical networks (HDNs) with coupling time-delays. Notions of input-to-state exponent, exponential ISS under events, and uniform exponential ISS under events are proposed. The exponential ISS under events and the uniform case are studied according to whether the flow gain is a small-gain or not. When the flow gain is a small-gain, Halanay-type ISS Lemmas are firstly established and used to derive criteria of exponential ISS under events and uniform exponential ISS under events. And the minimal dwell time between two consecutive events required for HDNs to achieve uniform exponential ISS under events is estimated. When the flow gain may be not a small-gain, by the method of Lyapunov-Krasovskii functional and M-matrix theory, criteria of exponential ISS under events and uniform exponential ISS under events are also established. And the maximal dwell time between two consecutive events is given for the uniform case. All conditions and results on the uniform case are easily checkable. Examples are given through the paper to illustrate the theoretical results.