An invariance principle for differential-algebraic equations with jumps

In this paper, we consider dynamical systems with multiple modes of operation and state jumps. Within each mode, the dynamics are given by linear differential-algebraic equations (DAEs). State jumps can occur when in a fixed mode as well as when transitioning between modes. We refer to this class of hybrid systems as hybrid DAEs. Motivated by the lack of results to study invariance properties of nonsmooth DAE systems, we characterize the properties of the omega limit set of solutions to these systems and propose an invariance principle. To this end, we employ results allowing for decomposition of DAEs (and switched DAEs) into the so-called quasi-Weierstrass form and for the study of invariance of hybrid inclusions. The results are illustrated in examples.