  This paper concerns a new class of discontinuous dynamical systems for
constrained optimization. These dynamics are particularly suited to solve
nonlinear, non-convex problems in closed-loop with a physical system. Such
approaches using feedback controllers that emulate optimization algorithms have
recently been proposed for the autonomous optimization of power systems and
other infrastructures. In this paper, we consider feedback gradient flows that
exploit physical input saturation with the help of anti-windup control to
enforce constraints. We prove semi-global convergence of "projected"
trajectories to first-order optimal points, i.e., of the trajectories obtained
from a pointwise projection onto the feasible set. In the process, we establish
properties of the directional derivative of the projection map for non-convex,
prox-regular sets.
