  This paper is concerned with the study of continuous-time, non-smooth
dynamical systems which arise in the context of time-varying non-convex
optimization problems, as for example the feedback-based optimization of power
systems. We generalize the notion of projected dynamical systems to
time-varying, possibly non-regular, domains and derive conditions for the
existence of so-called Krasovskii solutions. The key insight is that for
trajectories to exist, informally, the time-varying domain can only contract at
a bounded rate whereas it may expand discontinuously. This condition is met, in
particular, by feasible sets delimited via piecewise differentiable functions
under appropriate constraint qualifications. To illustrate the necessity and
usefulness of such a general framework, we consider a simple yet insightful
power system example, and we discuss the implications of the proposed
conditions for the design of feedback optimization schemes.
