In this paper, we propose a new control barrier function based quadratic program for general nonlinear control-affine systems, which, without any assumptions other than those taken in the original program, simultaneously guarantees forward invariance of the safety set, complete elimination of undesired equilibrium points inside it, and local asymptotic stability of the origin. To better appreciate this result, we first characterize the equilibrium points of the closed-loop system with the original quadratic program formulation. We then provide analytical results on how a certain parameter in the original quadratic program should be chosen to remove the undesired equilibrium points or to confine them in a small neighborhood of the origin. The new formulation then follows from these analytical results. Numerical examples are given alongside the theoretical discussions.
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