A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly

Abstract

We propose an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. These classes of systems are chosen due to their dominant prevalence in physics, mathematics, and control engineering research communities. A comparison of these frameworks is provided and their applicability depending on the character of the hybrid behavior is discussed. In particular, we show that special care should be taken when equations with Dirac delta function are interpreted as impulsive differential equations. We also provide insights on the stability and attractivity analysis of hybrid behaviors, highlight their essential differences to the respective stability concepts for smooth dynamical systems, and discuss specific phenomena which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques.