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

Universal symmetry of optimal control at the microscale