Optimal closed-loop control of active particles and a minimal information engine

We study the elementary problem of moving an active particle by a trap with minimum work input. We show analytically that (open-loop) optimal protocols are not affected by activity, but work fluctuations are always increased. For closed-loop protocols, which rely on initial measurements of the self-propulsion, the average work has a minimum for a finite persistence time. Using these insights, we derive an optimal periodic active information engine, which is found to have higher precision and information efficiency when operated with a run-and-tumble particle than for an active Ornstein-Uhlenbeck particle and, we argue, than for any other type of active particle.