  We propose a novel approach based on Denoising Diffusion Probabilistic Models
(DDPMs) to control nonlinear dynamical systems. DDPMs are the state-of-art of
generative models that have achieved success in a wide variety of sampling
tasks. In our framework, we pose the feedback control problem as a generative
task of drawing samples from a target set under control system constraints. The
forward process of DDPMs constructs trajectories originating from a target set
by adding noise. We learn to control a dynamical system in reverse such that
the terminal state belongs to the target set. For control-affine systems
without drift, we prove that the control system can exactly track the
trajectory of the forward process in reverse, whenever the the Lie bracket
based condition for controllability holds. We numerically study our approach on
various nonlinear systems and verify our theoretical results. We also conduct
numerical experiments for cases beyond our theoretical results on a
physics-engine.
