  This paper studies safety guarantees for systems with time-varying control
bounds. It has been shown that optimizing quadratic costs subject to state and
control constraints can be reduced to a sequence of Quadratic Programs (QPs)
using Control Barrier Functions (CBFs). One of the main challenges in this
method is that the CBF-based QP could easily become infeasible under tight
control bounds, especially when the control bounds are time-varying. The
recently proposed adaptive CBFs have addressed such infeasibility issues, but
require extensive and non-trivial hyperparameter tuning for the CBF-based QP
and may introduce overshooting control near the boundaries of safe sets. To
address these issues, we propose a new type of adaptive CBFs called
Auxiliary-Variable Adaptive CBFs (AVCBFs). Specifically, we introduce an
auxiliary variable that multiplies each CBF itself, and define dynamics for the
auxiliary variable to adapt it in constructing the corresponding CBF
constraint. In this way, we can improve the feasibility of the CBF-based QP
while avoiding extensive parameter tuning with non-overshooting control since
the formulation is identical to classical CBF methods. We demonstrate the
advantages of using AVCBFs and compare them with existing techniques on an
Adaptive Cruise Control (ACC) problem with time-varying control bounds.
