Universal symmetry of optimal control at the microscale

Optimal control is an important field in thermodynamics, mathematics and engineering which searches for strategies e.g. to move an object to a target position within a given time with minimal energetic effort. Especially for micro- and nanomachines, for which power supply is often limited, knowledge of optimal control protocols is crucial for their operation under realistic conditions. Here we investigate experimentally and theoretically the optimal protocol for transporting a microscopic colloidal particle with an optical trap in such a way that the required work is minimal. The experiments were conducted in viscous and viscoelastic media, which represent typical environments for synthetic and biological nanomachines. Despite marked differences between the protocols in both fluids, we find that the optimal transport protocol and the corresponding average particle trajectory always obeys time-reversal symmetry. This symmetry, which surprisingly appears here for a class of processes far away from thermal equilibrium, is moreover expected to hold universally for e.g. active, granular and long-range correlated media in their linear regimes. The uncovered symmetry provides a rigorous and versatile criterion for optimal control which greatly facilitates the search of energy-efficient transport strategies in a wide range of systems.