  Control Barrier Functions (CBFs) have become a popular tool for enforcing set
invariance in safety-critical control systems. While guaranteeing safety, most
CBF approaches are myopic in the sense that they solve an optimization problem
at each time step rather than over a long time horizon. This approach may allow
a system to get too close to the unsafe set where the optimization problem can
become infeasible. Some of these issues can be mitigated by introducing
relaxation variables into the optimization problem; however, this compromises
convergence to the desired equilibrium point. To address these challenges, we
develop an approximate optimal approach to the safety-critical control problem
in which the cost of violating safety constraints is directly embedded within
the value function. We show that our method is capable of guaranteeing both
safety and convergence to a desired equilibrium. Finally, we compare the
performance of our method with that of the traditional quadratic programming
approach through numerical examples.
