# Safe and Stable Control via Lyapunov-Guided Diffusion Models

arXiv: https://arxiv.org/abs/2509.25375 · HTML: https://mlsanity.com/paper/2509.25375/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2025.

## Built on by (most cited first, 10 of 10)

- [GDSD: Reinforcement Learning as Guided Denoiser Self-Distillation for Diffusion Language Models](https://mlsanity.com/paper/2605.29398/) (2026)
- [How Does the Lagrangian Guide Safe Reinforcement Learning through Diffusion Models?](https://mlsanity.com/paper/2602.02924/) (2026)
- [Planning with Language and Generative Models: Toward General Reward-Guided Wireless Network Design](https://mlsanity.com/paper/2602.00357/) (2026)
- [Fisher Decorator: Refining Flow Policy via a Local Transport Map](https://mlsanity.com/paper/2604.17919/) (2026)
- [MATT-Diff: Multimodal Active Target Tracking by Diffusion Policy](https://mlsanity.com/paper/2511.11931/) (2025)
- [Motion Planning with Model-Based Diffusion via Constraint Optimization and Adaptive Scheduling](https://mlsanity.com/paper/2607.14455/) (2026)
- [Coarse-Grained Kullback--Leibler Control of Diffusion-Based Generative AI](https://mlsanity.com/paper/2601.01045/) (2026)
- [Scaling World-Model Reinforcement Learning Through Diffusion Policy Optimization](https://mlsanity.com/paper/2605.26282/) (2026)
- [Let the Dynamics Flow: Stable Flow Matching Dynamical Systems](https://mlsanity.com/paper/2606.03834/) (2026)
- [Safe Score Matching: Diffusion Policies with Hamilton-Jacobi Reachability for Online Safe Reinforcement Learning](https://mlsanity.com/paper/2609.33337/) (2026)

## Abstract (arXiv)

Diffusion models have made significant strides in recent years, exhibiting strong generalization capabilities in planning and control tasks. However, most diffusion-based policies remain focused on reward maximization or cost minimization, often overlooking critical aspects of safety and stability. In this work, we propose Safe and Stable Diffusion (S2Diff), a model-based diffusion framework that explores how diffusion models can ensure safety and stability from a Lyapunov perspective. We demonstrate that S2Diff eliminates the reliance on both complex gradient-based solvers (e.g., quadratic programming, non-convex solvers) and control-affine structures, leading to globally valid control policies driven by the learned certificate functions. Additionally, we uncover intrinsic connections between diffusion sampling and Almost Lyapunov theory, enabling the use of trajectory-level control policies to learn better certificate functions for safety and stability guarantees. To validate our approach, we conduct experiments on a wide variety of dynamical control systems, where S2Diff consistently outperforms both certificate-based controllers and model-based diffusion baselines in terms of safety, stability, and overall control performance.