# Convex synthesis and verification of control-Lyapunov and barrier functions with input constraints

arXiv: https://arxiv.org/abs/2210.00629 · HTML: https://mlsanity.com/paper/2210.00629/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2022.

## Built on by (most cited first, 14 of 14)

- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
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- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (2023)
- [Verification of Neural Control Barrier Functions with Symbolic Derivative Bounds Propagation](https://mlsanity.com/paper/2410.16281/) (2024)
- [Safe Large-Scale Robust Nonlinear MPC in Milliseconds via Reachability-Constrained System Level Synthesis on the GPU](https://mlsanity.com/paper/2604.07644/) (2026)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control](https://mlsanity.com/paper/2411.18235/) (2024)
- [Formal Safety Verification and Refinement for Generative Motion Planners via Certified Local Stabilization](https://mlsanity.com/paper/2509.19688/) (2025)
- [Combining Neural Networks and Symbolic Regression for Analytical Lyapunov Function Discovery](https://mlsanity.com/paper/2406.15675/) (2024)
- [Analytical Lyapunov Function Discovery: An RL-based Generative Approach](https://mlsanity.com/paper/2502.02014/) (2025)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)
- [Provably Safe, Yet Scalable Reinforcement Learning](https://mlsanity.com/paper/2606.14536/) (2026)
- [Fault Tolerant Neural Control Barrier Functions for Robotic Systems under Sensor Faults and Attacks](https://mlsanity.com/paper/2402.18677/) (2024)

## Abstract (arXiv)

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) are widely used tools for synthesizing controllers subject to stability and safety constraints. Paired with online optimization, they provide stabilizing control actions that satisfy input constraints and avoid unsafe regions of state-space. Designing CLFs and CBFs with rigorous performance guarantees is computationally challenging. To certify existence of control actions, current techniques not only design a CLF/CBF, but also a nominal controller. This can make the synthesis task more expensive, and performance estimation more conservative. In this work, we characterize polynomial CLFs/CBFs using sum-of-squares conditions, which can be directly certified using convex optimization. This yields a CLF and CBF synthesis technique that does not rely on a nominal controller. We then present algorithms for iteratively enlarging estimates of the stabilizable and safe regions. We demonstrate our algorithms on a 2D toy system, a pendulum and a quadrotor.