# Off Policy Lyapunov Stability in Reinforcement Learning

arXiv: https://arxiv.org/abs/2509.09863 · HTML: https://mlsanity.com/paper/2509.09863/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 11 Sep 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 20 of 20)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Deep Reinforcement Learning for Robotics: A Survey of Real-World Successes](https://mlsanity.com/paper/2408.03539/) (2024)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Control of a Quadrotor with Reinforcement Learning](https://mlsanity.com/paper/1707.05110/) (2017)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [A Benchmark Comparison of Learned Control Policies for Agile Quadrotor Flight](https://mlsanity.com/paper/2202.10796/) (2022)
- [Actor-Critic Reinforcement Learning for Control with Stability Guarantee](https://mlsanity.com/paper/2004.14288/) (2020)
- [Stabilizing Neural Control Using Self-Learned Almost Lyapunov Critics](https://mlsanity.com/paper/2107.04989/) (2021)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Refining Control Barrier Functions through Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2204.12507/) (2022)
- [Stable and Safe Reinforcement Learning via a Barrier-Lyapunov Actor-Critic Approach](https://mlsanity.com/paper/2304.04066/) (2023)
- [Reinforcement Learning for Safe Robot Control using Control Lyapunov Barrier Functions](https://mlsanity.com/paper/2305.09793/) (2023)
- [Model-free Neural Lyapunov Control for Safe Robot Navigation](https://mlsanity.com/paper/2203.01190/) (2022)
- [Critic as Lyapunov function (CALF): a model-free, stability-ensuring agent](https://mlsanity.com/paper/2409.09869/) (2024)

## Built on by (most cited first, 2 of 2)

- [LC-SAC: Lyapunov-Constrained Soft Actor-Critic via Koopman Operator Theory for Trajectory Tracking and Stabilization](https://mlsanity.com/paper/2602.04132/) (2026)
- [When Does External Guidance Help LLM Reasoning? A Bias-Variance Theory of Guidance-Augmented GRPO](https://mlsanity.com/paper/2610.06861/) (2026)

## Abstract (arXiv)

Traditional reinforcement learning lacks the ability to provide stability guarantees. More recent algorithms learn Lyapunov functions alongside the control policies to ensure stable learning. However, the current self-learned Lyapunov functions are sample inefficient due to their on-policy nature. This paper introduces a method for learning Lyapunov functions off-policy and incorporates the proposed off-policy Lyapunov function into the Soft Actor Critic and Proximal Policy Optimization algorithms to provide them with a data efficient stability certificate. Simulations of an inverted pendulum and a quadrotor illustrate the improved performance of the two algorithms when endowed with the proposed off-policy Lyapunov function.