  Safety is one of the fundamental problems in robotics. Recently, a quadratic
program-based control barrier function (CBF) method has emerged as a way to
enforce safety-critical constraints. Together with control Lyapunov function
(CLF), it forms a safety-critical control strategy, named CLF-CBF-QP, which can
mediate between achieving the control objective and ensuring safety, while
being executable in real-time. However, once additional constraints such as
input constraints are introduced, the CLF-CBF-QP may encounter infeasibility.
In order to address the challenge that arises due to the infeasibility, we
propose an optimal-decay form for safety-critical control wherein the decay
rate of the CBF is optimized point-wise in time so as to guarantee point-wise
feasibility when the state lies inside the safe set. The proposed control
design is numerically validated using an adaptive cruise control example.
