# Permissive Barrier Certificates for Safe Stabilization Using Sum-of-squares

arXiv: https://arxiv.org/abs/1802.08917 · HTML: https://mlsanity.com/paper/1802.08917/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2018.

## Built on by (most cited first, 15 of 15)

- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Gameplay Filters: Robust Zero-Shot Safety through Adversarial Imagination](https://mlsanity.com/paper/2405.00846/) (2024)
- [Learning Robust Hybrid Control Barrier Functions for Uncertain Systems](https://mlsanity.com/paper/2101.06492/) (2021)
- [Deception Game: Closing the Safety-Learning Loop in Interactive Robot Autonomy](https://mlsanity.com/paper/2309.01267/) (2023)
- [Reactive and Safe Road User Simulations using Neural Barrier Certificates](https://mlsanity.com/paper/2109.06689/) (2021)
- [System-level Safety Guard: Safe Tracking Control through Uncertain Neural Network Dynamics Models](https://mlsanity.com/paper/2312.06810/) (2023)
- [Model Free Barrier Functions via Implicit Evading Maneuvers](https://mlsanity.com/paper/2107.12871/) (2021)
- [ConBaT: Control Barrier Transformer for Safe Policy Learning](https://mlsanity.com/paper/2303.04212/) (2023)
- [Safe Reinforcement Learning for Power System Control: A Review](https://mlsanity.com/paper/2407.00681/) (2024)
- [Barrier Certified Safety Learning Control: When Sum-of-Square Programming Meets Reinforcement Learning](https://mlsanity.com/paper/2206.07915/) (2022)
- [Pre-Trained Vision Models as Perception Backbones for Safety Filters in Autonomous Driving](https://mlsanity.com/paper/2410.22585/) (2024)
- [Data-Driven Safety Verification using Barrier Certificates and Matrix Zonotopes](https://mlsanity.com/paper/2504.01007/) (2025)
- [Learning Neural Control Barrier Functions from Expert Demonstrations using Inverse Constraint Learning](https://mlsanity.com/paper/2510.21560/) (2025)
- [A Barrier Certificate-based Simplex Architecture for Systems with Approximate and Hybrid Dynamics](https://mlsanity.com/paper/2202.09710/) (2022)

## Abstract (arXiv)

Motivated by the need to simultaneously guarantee safety and stability of safety-critical dynamical systems, we construct permissive barrier certificates in this paper that explicitly maximize the region where the system can be stabilized without violating safety constraints. An optimization strategy is developed to search for the maximum volume barrier certified region of safe stabilization. The barrier certified region, which is allowed to take any arbitrary shape, is proved to be strictly larger than safe regions generated with Lyapunov sublevel set based methods. The proposed approach effectively unites a Lyapunov function with multiple barrier functions that might not be compatible with each other. Iterative search algorithms are developed using sum-of-squares to compute the most permissive, that is, the maximum volume, barrier certificates. Simulation results of the iterative search algorithm demonstrate the effectiveness of the proposed method.