Finite-time transitions in optimal control and non-equilibrium relaxation

How should one steer a small particle through a noisy and inhomogeneous environment when certain spatial regions carry additional energetic costs? We study this question for a colloidal particle dragged by optical tweezers towards an obstacle with spatially varying penalty. The optimal protocol has to balance viscous dissipation along the path against the energetic penalty at the end. This competition produces a sharp transition in the control strategy at a critical protocol duration. For short durations, it is best to accept the penalty and not steer at all. For long durations, the optimal protocol drives the particle towards a low-cost region. For symmetric obstacles the choice of protocol breaks the symmetry spontaneously, much as in a continuous phase transition.

Stochastic optimal control is closely related to large deviation theory. Using this connection, we map the control cost onto the rate function of rare trajectories in relaxation after a quench. The control transition then becomes a finite-time dynamical phase transition. Typically, sampling such transitions directly is exponentially costly, but the control experiment obtains them through ordinary averages. We observe both transitions in experiments with optically trapped colloids.

In the last part of the talk I show that these results carry over to general quadratic control problems. These problems fall into three equivalence classes known from classical mechanics. Only two of the three have a simple counterpart in relaxation.

The talk is based on these works on arXiv:

• Finite-time transitions in optimal control and non-equilibrium relaxation
• Equivalence classes of finite-time transitions in optimal control and non-equilibrium relaxation