Abstract: Knowing the protocol by which a system can be brought from one state to another in such a way that the energy input or entropy generated is minimized is useful for building efficient machines or understanding biological processes. Such optimal protocols can be highly counterintutive, even for simple processes with no memory or in-build activity [1]. We consider the canonical control problem of moving a harmonic trap that contains a Brownian particle over a given distance in a given time using as little work as possible. For viscous media, it is known that the optimal protocol has sudden jumps at the beginning and at the end and is characterized by a constant power input in between [1]. However, in various real-world scenarios, e.g. subcellular transport processes, the assumption of a purely visous medium is not justified. We show that memory effects in the environment lead to significantly different optimal strategies where the trap velocity needs to be continuously modulated [2]. Surprisingly, for all linear media the optimal solutions still exhibit a universal symmetry, regardless of the memory properties of the environment. We present experimental evidence for our findings using a colloidal particle dragged by an optimal trap through viscous and viscoelastic fluids. Finally, I will give an outlook on ongoing research on the optimal control of active particles.

[1] T. Schmiedl and U. Seifert, PRL 98, 108301, (2007).
[2] S.A.M. Loos, S. Monter, F. Ginot, C. Bechinger, Universal symmetry of optimal control at the microscale, ArXiv 2311.00470 (2023).