Oberseminar "Dynamische Systeme und Kontrolltheorie" - Patrick Bachmann

Impulsive systems are a class of hybrid dynamical systems which combine continuous behavior with sudden changes of state. Impulsive systems have a large field of applications such as biomedicine, networked control systems, robotics, and neural networks. In practice, an important property of dynamical systems is their sensitivity to external perturbations. To categorize the effects of these perturbations, the notion of input-to-state stability (ISS) was introduced which allows to identify systems with a certain tolerance to perturbations.

In our talk, we consider impulsive systems over Banach spaces and present ISS-Lyapunov functions which not only determine a sufficient but also a necessary condition for input-to-state stability. This is achieved by the application of time-varying Lyapunov functions. Compared to candidate ISS-Lyapunov functions, our method allows to conclude stability for a broader class of impulsive systems, especially for systems which possess unstable continuous and unstable discrete dynamics coincidentally. Furthermore, we give stochastic ISS conditions for a class of impulsive systems with random jump times.