On necessary and sufficient conditions for incremental stability of hybrid systems using the graphical distance between solutions

This paper introduces new incremental stability notions for a class of hybrid dynamical systems given in terms of differential equations and difference equations with state constraints. Incremental stability is defined as the property that the distance between every pair of solutions to the system has stable behavior (incremental stability) and approaches zero asymptotically (incremental attractivity) in terms of graphical convergence. Basic properties of the class of graphically incrementally stable systems are considered as well as those implied by the new notions are revealed. Moreover, several sufficient and necessary conditions for a hybrid system with such a property are established. Examples are presented throughout the paper to illustrate the notions and results.