# Approximate Optimal Control for Safety-Critical Systems with Control Barrier Functions

arXiv: https://arxiv.org/abs/2008.04122 · HTML: https://mlsanity.com/paper/2008.04122/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2020.

## Built on by (most cited first, 6 of 6)

- [Safe Exploration in Model-based Reinforcement Learning using Control Barrier Functions](https://mlsanity.com/paper/2104.08171/) (2021)
- [Differentiable Safe Controller Design through Control Barrier Functions](https://mlsanity.com/paper/2209.10034/) (2022)
- [Neuro-Symbolic Safety Guidance for Vision-Language-Action Models via Constrained Flow Matching](https://mlsanity.com/paper/2607.01378/) (2026)
- [Lyapunov Neural ODE State-Feedback Control Policies](https://mlsanity.com/paper/2409.00393/) (2024)
- [Uncertainty Aware-Predictive Control Barrier Functions: Safer Human Robot Interaction through Probabilistic Motion Forecasting](https://mlsanity.com/paper/2508.20812/) (2025)
- [Autopilot-Preserving Residual Q-Learning with HJB-Inspired Finite-Action Risk Filtering for Fixed-Wing UAV Command Supervision](https://mlsanity.com/paper/2606.01397/) (2026)

## Abstract (arXiv)

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.