# Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control

**Authors:** Yiting Chen, Pol Mestres, Jorge Cortes, Emiliano Dall'Anese  
**arXiv:** [2409.06808](https://arxiv.org/abs/2409.06808)  
**Date:** 2024-09-10  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2409.06808v1/

## Key points

- Control Barrier Functions (CBFs) are crucial for ensuring safety in optimization-based control for control-affine systems
- CBFs are associated with a designated "safe" set and integrated into the control law optimization problem to maintain forward invariance of the safe set and ensure system safety
- A challenge arises with CBF-based control laws introducing undesirable equilibrium points that do not align with the original system's equilibria
- Different choices of CBFs affect closed-loop system dynamics, but do not alter equilibria within the safe set's interior, only manifesting along its boundary
- The number and placement of undesirable equilibria within closed-loop systems remain consistent regardless of CBF selection
- Stability properties are not influenced by variations in CBF selection or associated extended class-K functions

## Summary

In the realm of safe optimization-based control for control-affine systems, Control Barrier Functions (CBFs) play a crucial role in ensuring the safety of controllers. These CBFs are associated with a designated "safe" set and are incorporated into the optimization problem that defines the control law to uphold the forward invariance of the safe set and guarantee safety within the system. However, a notable challenge arises with the introduction of undesirable equilibrium points by the CBF-based control law – these points do not align with the original system's equilibria. A recent study delves into exploring how different choices of CBFs can affect closed-loop system dynamics and reveals several key findings: CBFs do not alter equilibria within the safe set's interior but exclusively manifest along its boundary; there is no variation in either number or placement of undesirable equilibria within closed-loop systems regardless of CBF selection; stability properties remain unaffected by variations in CBF selection or associated extended class-K functions. This research highlights how nuanced decisions regarding CBFs can impact system dynamics while affirming certain stability properties within closed-loop systems regardless of specific CBF selections.