Work Extraction via Backward Motion in Optimal Closed-Loop Stochastic Control

We experimentally realize finite-time feedback control in an overdamped colloidal system using real-time optical tweezers with in situ reinforcement learning (RL). By varying the protocol duration tf for displacing the optical trap between prescribed positions, the optimal strategies identified by RL reveal a crossover from deterministic dragging toward the target to feedback-assisted exploitation of thermal fluctuations, reducing and eventually overcoming the energetic cost. The resulting policies agree quantitatively with the exact optimal closed-loop solution. By extending the approach to spatially localized external forcing, we further show that RL can identify optimal feedback strategies in heterogeneous stochastic environments where direct analytical control design is challenging.