  This paper studies the dynamical properties of closed-loop systems obtained
from control barrier function-based safety filters. We provide a sufficient and
necessary condition for the existence of undesirable equilibria and show that
the Jacobian matrix of the closed-loop system evaluated at an undesirable
equilibrium always has a nonpositive eigenvalue. In the special case of linear
planar systems and ellipsoidal obstacles, we give a complete characterization
of the dynamical properties of the corresponding closed-loop system. We show
that for underactuated systems, the safety filter always introduces a single
undesired equilibrium, which is a saddle point. We prove that all trajectories
outside the global stable manifold of such equilibrium converge to the origin.
In the fully actuated case, we discuss how the choice of nominal controller
affects the stability properties of the closed-loop system. Various simulations
illustrate our results.
