A Time-Reversal Control Synthesis for Steering the State of Stochastic Systems

This paper presents a novel approach for steering the state of a stochastic control-affine system to a desired target within a finite time horizon. Our method leverages the time-reversal of diffusion processes to construct the required feedback control law. Specifically, the control law is the so-called score function associated with the time-reversal of random state trajectories that are initialized at the target state and are simulated backwards in time. A neural network is trained to approximate the score function, enabling applicability to both linear and nonlinear stochastic systems. Numerical experiments demonstrate the effectiveness of the proposed method across several benchmark examples.