# Neural Network-assisted Interval Reachability for Systems with Control Barrier Function-Based Safe Controllers

arXiv: https://arxiv.org/abs/2504.08249 · HTML: https://mlsanity.com/paper/2504.08249/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 50 references that are in the map.
- First posted on arXiv 11 Apr 2025.

## Builds on (most cited first, 30 of 30)

- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Reach-Avoid Problems with Time-Varying Dynamics, Targets and Constraints](https://mlsanity.com/paper/1410.6445/) (2014)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems](https://mlsanity.com/paper/2106.13867/) (2021)
- [OVERT: An Algorithm for Safety Verification of Neural Network Control Policies for Nonlinear Systems](https://mlsanity.com/paper/2108.01220/) (2021)
- [Reachable Set Estimation for Neural Network Control Systems: A Simulation-Guided Approach](https://mlsanity.com/paper/2004.12273/) (2020)
- [Control Barrier Functions for Systems with High Relative Degree](https://mlsanity.com/paper/1903.04706/) (2019)
- [Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes](https://mlsanity.com/paper/2112.09197/) (2021)
- [Refining Control Barrier Functions through Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2204.12507/) (2022)
- [Safe Control Synthesis via Input Constrained Control Barrier Functions](https://mlsanity.com/paper/2104.01704/) (2021)
- [Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation](https://mlsanity.com/paper/2305.18460/) (2023)
- [Interval Reachability of Nonlinear Dynamical Systems with Neural Network Controllers](https://mlsanity.com/paper/2301.07912/) (2023)
- [Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loops](https://mlsanity.com/paper/2307.14938/) (2023)
- [A Robust, Multiple Control Barrier Function Framework for Input Constrained Systems](https://mlsanity.com/paper/2205.13726/) (2022)
- [Advances in the Theory of Control Barrier Functions: Addressing Practical Challenges in Safe Control Synthesis for Autonomous and Robotic Systems](https://mlsanity.com/paper/2312.16719/) (2023)
- [Fast, Smooth, and Safe: Implicit Control Barrier Functions through Reach-Avoid Differential Dynamic Programming](https://mlsanity.com/paper/2307.00193/) (2023)
- [On the Parameterized Computation of Minimum Volume Outer Ellipsoid of Minkowski Sum of Ellipsoids](https://mlsanity.com/paper/1803.08157/) (2018)
- [Reach-Avoid Problems via Sum-of-Squares Optimization and Dynamic Programming](https://mlsanity.com/paper/1807.11553/) (2018)
- [Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria](https://mlsanity.com/paper/2003.07819/) (2020)
- [On the Undesired Equilibria Induced by Control Barrier Function Based Quadratic Programs](https://mlsanity.com/paper/2104.14895/) (2021)
- [Continuous Reachability Task Transition Using Control Barrier Functions](https://mlsanity.com/paper/2005.12407/) (2020)
- [Enforcing Safety at Runtime for Systems with Disturbances](https://mlsanity.com/paper/2008.07019/) (2020)
- [On Compatibility and Region of Attraction for Safe, Stabilizing Control Laws](https://mlsanity.com/paper/2008.12179/) (2020)
- [Characterization of the Dynamical Properties of Safety Filters for Linear Planar Systems](https://mlsanity.com/paper/2408.00958/) (2024)
- [Independence of Closed-Loop Equilibria and Stability from the Choice of Control Barrier Function for a Given Safe Set](https://mlsanity.com/paper/2409.06808/) (2024)

## Abstract (arXiv)

Control Barrier Functions (CBFs) have been widely utilized in the design of optimization-based controllers and filters for dynamical systems to ensure forward invariance of a given set of safe states. While CBF-based controllers offer safety guarantees, they can compromise the performance of the system, leading to undesirable behaviors such as unbounded trajectories and emergence of locally stable spurious equilibria. Computing reachable sets for systems with CBF-based controllers is an effective approach for runtime performance and stability verification, and can potentially serve as a tool for trajectory re-planning. In this paper, we propose a computationally efficient interval reachability method for performance verification of systems with optimization-based controllers by: (i) approximating the optimization-based controller by a pre-trained neural network to avoid solving optimization problems repeatedly, and (ii) using mixed monotone theory to construct an embedding system that leverages state-of-the-art neural network verification algorithms for bounding the output of the neural network. Results in terms of closeness of solutions of trajectories of the system with the optimization-based controller and the neural network are derived. Using a single trajectory of the embedding system along with our closeness of solutions result, we obtain an over-approximation of the reachable set of the system with optimization-based controllers. Numerical results are presented to corroborate the technical findings.