Incremental graphical asymptotic stability for hybrid dynamical systems

Abstract

This chapter introduces an incremental asymptotic stability notion for sets of hybrid trajectories J. The elements in J are functions defined on hybrid time domains, which are subsets of R≥0xN with a specific structure. For this abstract system, incremental asymptotic stability is defined as the property of the graphical distance between every pair of solutions to the system having stable behavior (incremental graphical stability) and approaching zero asymptotically (incremental graphical attractivity). Necessary conditions for J to have such properties are presented. When J is generated by hybrid systems given in terms of hybrid inclusions, that is, differential equations and difference equations with state constraints, further necessary conditions on the data are highlighted. In addition, sufficient conditions for incremental graphical asymptotic stability involving the data of the hybrid inclusion are presented. Throughout the chapter, examples illustrate the notions and results.