Safe and Stable Control via Guided Diffusion Models

Diffusion models have made significant strides in recent years, exhibiting strong generalization capabilities in planning and control tasks. However, most diffusionbased 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 gradientbased solvers (e.g., quadratic programming, non-convex solvers) and controlaffine 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 trajectorylevel 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 certificatebased controllers and model-based diffusion baselines in terms of safety, stability, and overall control performance.