Safe and Stable Control via Lyapunov-Guided Diffusion Models (2509.25375v1)

Abstract: 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 ($S^2$Diff), a model-based diffusion framework that explores how diffusion models can ensure safety and stability from a Lyapunov perspective. We demonstrate that $S^2$Diff 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 $S^2$Diff consistently outperforms both certificate-based controllers and model-based diffusion baselines in terms of safety, stability, and overall control performance.

• The paper introduces S²Diff, a framework that integrates Lyapunov functions with diffusion models to ensure safe and stable control.
• It reformulates control as a probabilistic sampling problem, reducing rejection rates and enabling robust convergence in complex, non-convex dynamics.
• Experimental results show lower oscillations and reduced tracking errors compared to traditional QP methods and MPC, highlighting its practical efficiency.

Safe and Stable Control via Lyapunov-Guided Diffusion Models

Introduction

The paper presents an approach for enhancing safety and stability in control systems using diffusion models guided by Lyapunov functions. While diffusion models have shown robust performance in planning and control, challenges persist in ensuring safety and stability, often essential in applications involving robotics and aerospace. Existing methods typically rely on complex QP-based solvers or constraints, leading to inefficiencies when dealing with high-dimensional non-convex problems. The approach introduced, termed Safe and Stable Diffusion (S^2Diff), integrates Almost Lyapunov theory and diffusion sampling to learn control policies that inherently satisfy these requirements by leveraging the properties of control Lyapunov barrier functions (CLBFs).

Probabilistic Formulation and Diffusion Sampling

S^2Diff reinterprets the control problem as a sampling task from a probabilistic distribution designed to prioritize safety, stability, and cost-efficiency. It avoids traditional constraints by framing the problem probabilistically, allowing for trajectory sampling that emphasizes the almost sure convergence of stability guaranteed by CLBFs. This probabilistic approach mitigates the high rejection rates associated with strict enforcement of constraints, targeting global safety and stability through an energy parameterization that emphasizes trajectory-level control effectiveness over almost Lyapunov stability.

Figure 1: Benchmark control tasks for safety and stability.

CLBF Update via Sampled Trajectories

The method iteratively refines CLBFs using data from diffusion-sampled trajectories. The update process employs a loss function crafted to ensure that each parameterized CLBF maintains equilibrium properties while obeying safety and stability constraints, even under non-convex dynamics without requiring explicit control-affine assumptions. Utilizing automatic differentiation, the approach effectively scales to real-world systems by adapting both continuous-time and discrete-time Lyapunov conditions.

Theoretical Insights and Guarantees

The paper establishes theoretical grounds by employing Almost Lyapunov theory to demonstrate safety and stability under minor probabilistic violations, ensuring that the CLBF-guided diffusion policies converge with a high degree of certainty. The Almost Lyapunov framework delivers exponential decay in the Lyapunov function values along trajectories, proving that sampled policies maintain effective global convergence despite local deviations. Furthermore, the analysis provides robustness measurements indicating that learned certificate functions minimize violation regions to negligible levels within compact state spaces.

Experimental Results

Experiments conducted across benchmark systems, such as inverted pendulums and aircraft control scenarios, highlight S^2Diff's capability to outperform model-based diffusion approaches and QP methods regarding safety and stability metrics. Empirically, the sampling efficiency and policy consistency are reinforced by lower oscillations and reduced tracking errors across control tasks, solidifying the practical applicability of the method in complex dynamic environments.

Figure 2: Left: CLBFs learned by Gradient-based method (left-1) vs. Diffusion Sampling (left-2) for inverted pendulum. Right: Contour maps along different axes of the CLBF learned by S^2Diff for the high-dimensional, non-control-affine F-16 with non-convex constraints. The smooth level sets across 2D projections highlight the CLBF’s expressiveness and its ability to capture complex, constrained dynamics.

Ablation Studies

Ablation studies further explore S^2Diff's robustness to hyperparameters such as trajectory length and stability temperature. Results suggest optimal settings that balance safety, stability, and computational cost, providing insights into tuning the framework for varied real-world applications. Moreover, comparisons against direct gradient-based and MPC methods illustrate S^2Diff's advantages in efficiency and effectiveness across different model settings.

Figure 3: Control trajectories of a 2D quadrotor with four methods including ours (S^2Diff). The circ and times mark the start and end points, respectively. Green lines denote safe states; red lines indicate constraint violations. S^2Diff achieves higher safety and stability, effectively handling non-convex constraints where baselines struggle.

Conclusion

S^2Diff innovatively bridges diffusion models with Lyapunov-guided control, offering enhanced safety and stability in dynamic systems without requiring explicit slack variables or affine constraints. Despite the promising performance, future efforts can explore improvements in diffusion speed and real-world deployment scenarios. The methodological balance between theoretical rigor and practical applicability underscores the potential for S^2Diff to advance the reliability of autonomous systems across diverse control domains.