# Learning Control Lyapunov Functions from Counterexamples and Demonstrations

arXiv: https://arxiv.org/abs/1804.05285 · HTML: https://mlsanity.com/paper/1804.05285/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2018.

## Built on by (most cited first, 19 of 19)

- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Stability Verification in Stochastic Control Systems via Neural Network Supermartingales](https://mlsanity.com/paper/2112.09495/) (2021)
- [Neural Lyapunov Control for Discrete-Time Systems](https://mlsanity.com/paper/2305.06547/) (2023)
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- [Neural Koopman Lyapunov Control](https://mlsanity.com/paper/2201.05098/) (2022)
- [Neural Abstractions](https://mlsanity.com/paper/2301.11683/) (2023)
- [Verification-Aided Learning of Neural Network Barrier Functions with Termination Guarantees](https://mlsanity.com/paper/2403.07308/) (2024)
- [A General Framework for Verification and Control of Dynamical Models via Certificate Synthesis](https://mlsanity.com/paper/2309.06090/) (2023)
- [Learning Stability Attention in Vision-based End-to-end Driving Policies](https://mlsanity.com/paper/2304.02733/) (2023)
- [Adversarially Robust Stability Certificates can be Sample-Efficient](https://mlsanity.com/paper/2112.10690/) (2021)
- [Formal Policy Learning from Demonstrations for Reachability Properties](https://mlsanity.com/paper/1903.00589/) (2019)
- [Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Control](https://mlsanity.com/paper/2201.00801/) (2022)
- [Enabling Efficient, Reliable Real-World Reinforcement Learning with Approximate Physics-Based Models](https://mlsanity.com/paper/2307.08168/) (2023)
- [A Guaranteed-Stable Neural Network Approach for Optimal Control of Nonlinear Systems](https://mlsanity.com/paper/2501.17333/) (2025)
- [Provably-Stable Neural Network-Based Control of Nonlinear Systems](https://mlsanity.com/paper/2502.00248/) (2025)
- [Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks](https://mlsanity.com/paper/2504.02607/) (2025)
- [Reinforcement Learning for Robust Ageing-Aware Control of Li-ion Battery Systems with Data-Driven Formal Verification](https://mlsanity.com/paper/2509.04288/) (2025)
- [Lyapunov Stability Learning with Nonlinear Control via Inductive Biases](https://mlsanity.com/paper/2511.01283/) (2025)

## Abstract (arXiv)

We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function, and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.