Gain estimates and exponential ISS for a class of hybrid dynamical networks

This paper studies the exponential input-to-state stability (e-ISS) for hybrid dynamical networks (HDNs) consisting of flow and jumping subsystems. Concepts of input-to-state exponent (IS-E) and event-ISS are proposed. By using the IS-E estimates of subsystems and the method of multiple Lyapunov functions and M-matrix theory, the e-ISS including event-e-ISS criteria are established for HDNs. The small-gain condition for the coupling of flows is needed while such condition for the coupling of impulses is not necessary. It also shows that a HDN may have ISS even all subsystems have no ISS. Finally, one example is given to illustrate the results.