Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems

In this paper, we propose a deterministic diffusion-based framework for controlling the probability density of nonlinear control-affine systems, with theoretical guarantees for drift-free and linear time-invariant (LTI) dynamics. The central idea is to first excite the system with white noise so that a forward diffusion process explores the reachable regions of state space, and then to design a deterministic feedback law that acts as a denoising mechanism driving the system back toward a desired target distribution supported on the target set. This denoising phase provides a feedback controller that steers the control system to the target set. In this framework, control synthesis reduces to constructing a deterministic reverse process that reproduces the desired evolution of state densities. We derive existence conditions ensuring such deterministic realizations of time-reversals for controllable drift-free and LTI systems, and show that the resulting feedback laws provide a tractable alternative to nonlinear control by viewing density control as a relaxation of controlling a system to target sets. Numerical studies on a unicycle model with obstacles, a five-dimensional driftless system, and a four-dimensional LTI system demonstrate reliable diffusion-inspired density control.