Input-to-state stability for a class of hybrid dynamical systems via hybrid time approach

This paper studies the ISS (input-to-state stability) for a class of HDS (hybrid dynamical systems). By using the concept of hybrid time for HDS, two kinds of ISS notions are proposed. They are called the first ISS property and the second ISS property of HDS. By employing the ISS properties on continuous and/or discrete dynamics in the HDS, the first and second ISS properties for the whole HDS are investigated. We show that there exists the first ISS property for the whole HDS if any one of dynamics in the HDS has the ISS property and the dwell time on this dynamics is sufficiently long even if there is no ISS property for the other dynamics. Moreover, the second ISS property of HDS is derived for the cases in which one kind of dynamics without external inputs has a stability property while the other one has no stability and ISS properties. Finally, one example is given for illustration.