# Learning Lyapunov Functions for Hybrid Systems

arXiv: https://arxiv.org/abs/2012.12015 · HTML: https://mlsanity.com/paper/2012.12015/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2020.

## Built on by (most cited first, 24 of 24)

- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Neural Lyapunov Control for Discrete-Time Systems](https://mlsanity.com/paper/2305.06547/) (2023)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [Neural Termination Analysis](https://mlsanity.com/paper/2102.03824/) (2021)
- [Learning Robust Hybrid Control Barrier Functions for Uncertain Systems](https://mlsanity.com/paper/2101.06492/) (2021)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [Stability Verification of Neural Network Controllers using Mixed-Integer Programming](https://mlsanity.com/paper/2206.13374/) (2022)
- [Neural Abstractions](https://mlsanity.com/paper/2301.11683/) (2023)
- [LyZNet: A Lightweight Python Tool for Learning and Verifying Neural Lyapunov Functions and Regions of Attraction](https://mlsanity.com/paper/2403.10013/) (2024)
- [Provably Safe Neural Network Controllers via Differential Dynamic Logic](https://mlsanity.com/paper/2402.10998/) (2024)
- [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)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Provably Correct Training of Neural Network Controllers Using Reachability Analysis](https://mlsanity.com/paper/2102.10806/) (2021)
- [Neural Continuous-Time Supermartingale Certificates](https://mlsanity.com/paper/2412.17432/) (2024)
- [Learning Contraction Policies from Offline Data](https://mlsanity.com/paper/2112.05911/) (2021)
- [Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control](https://mlsanity.com/paper/2411.18235/) (2024)
- [Model-free Learning of Regions of Attraction via Recurrent Sets](https://mlsanity.com/paper/2204.10372/) (2022)
- [Hybrid Systems Neural Control with Region-of-Attraction Planner](https://mlsanity.com/paper/2303.10327/) (2023)
- [MSACL: Multi-Step Actor-Critic Learning with Lyapunov Certificates for Exponentially Stabilizing Control](https://mlsanity.com/paper/2512.24955/) (2025)

## Abstract (arXiv)

We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical system in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to find a valid Lyapunov function inside a convex set of Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.