Discrete Abstractions of Infinite-Dimensional Impulsive Systems

We provide discrete abstractions of impulsive systems on Banach spaces. Thereby we explicitly allow infinite-dimensional state and input spaces, which makes it possible to cover a crucially important class of dynamical systems modeled by partial differential equations with jumps, referred to as impulsive evolution systems. Using the notion of the so-called alternating simulation function, we prove, under an incremental stability assumption, that there exists an approximate alternating simulation relation between the impulsive system and its discrete abstraction. We also provide conditions for the existence of an approximate alternating bisimulation relation. A notable feature of our work is that we propose a time-varying alternating simulation function that allows the construction of discrete abstractions for a broad class of impulsive systems in which both the flow and jumps are possibly unstable. Our method also covers the classes of time-varying impulsive systems and impulsive systems with an output map.