# Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

arXiv: https://arxiv.org/abs/2003.07819 · HTML: https://mlsanity.com/paper/2003.07819/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Mar 2020.

## Built on by (most cited first, 3 of 3)

- [Discrete GCBF Proximal Policy Optimization for Multi-agent Safe Optimal Control](https://mlsanity.com/paper/2502.03640/) (2025)
- [Neural Network-assisted Interval Reachability for Systems with Control Barrier Function-Based Safe Controllers](https://mlsanity.com/paper/2504.08249/) (2025)
- [Risk-Aware Safe Reinforcement Learning for Control of Stochastic Linear Systems](https://mlsanity.com/paper/2505.09734/) (2025)

## Abstract (arXiv)

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs), guaranteeing safety in the form of trajectory invariance with respect to a given set. In this manuscript, we show that this framework can introduce equilibrium points (particularly at the boundary of the unsafe set) other than the minimum of the Lyapunov function into the closed-loop system. We derive explicit conditions under which these undesired equilibria (which can even appear in the simple case of linear systems with just one convex unsafe set) are asymptotically stable. To address this issue, we propose an extension to the QP-based controller unifying CLFs and CBFs that explicitly avoids undesirable equilibria on the boundary of the safe set. The solution is illustrated in the design of a collision-free controller.