  It has been shown that satisfying state and control constraints while
optimizing quadratic costs subject to desired (sets of) state convergence for
affine control systems can be reduced to a sequence of quadratic programs (QPs)
by using Control Barrier Functions (CBFs) and Control Lyapunov Functions
(CLFs). One of the main challenges in this approach is ensuring the feasibility
of these QPs, especially under tight control bounds and safety constraints of
high relative degree. In this paper, we provide sufficient conditions for
guranteed feasibility. The sufficient conditions are captured by a single
constraint that is enforced by a CBF, which is added to the QPs such that their
feasibility is always guaranteed. The additional constraint is designed to be
always compatible with the existing constraints, therefore, it cannot make a
feasible set of constraints infeasible - it can only increase the overall
feasibility. We illustrate the effectiveness of the proposed approach on an
adaptive cruise control problem.
