  Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have
been used to develop provably safe controllers by means of quadratic programs
(QPs), guaranteeing safety in the form of trajectory invariance with respect to
a given set. In this manuscript, we show that this framework can introduce
equilibrium points (particularly at the boundary of the unsafe set) other than
the minimum of the Lyapunov function into the closed-loop system. We derive
explicit conditions under which these undesired equilibria (which can even
appear in the simple case of linear systems with just one convex unsafe set)
are asymptotically stable. To address this issue, we propose an extension to
the QP-based controller unifying CLFs and CBFs that explicitly avoids
undesirable equilibria on the boundary of the safe set. The solution is
illustrated in the design of a collision-free controller.
