Active particles in moving traps: Minimum work protocols and information efficiency of work extraction

We revisit the elementary problem of moving a particle in a harmonic trap in finite time with minimal work cost and extend it to the case of an active particle. By comparing the Gaussian case of an Active Ornstein-Uhlenbeck particle and the non-Gaussian run-and-tumble particle, we establish general principles for thermodynamically optimal control of active matter beyond specific models. We show that the open-loop optimal protocols, which do not incorporate system-state information, are identical to those of passive particles but result in larger work fluctuations due to activity. In contrast, closed-loop (or feedback) control with a single (initial) measurement changes the optimal protocol and reduces the average work relative to the open-loop control for small enough measurement errors. Minimum work is achieved by particles with finite persistence time. As an application, we propose an active information engine which extracts work from self-propulsion. This periodic engine achieves higher information efficiency with run-and-tumble particles than with active Ornstein-Uhlenbeck particles. Complementing a companion paper that gives only the main results [Garcia-Millan et al., companion paper, Phys. Rev. Lett. 135, 088301 (2025)10.1103/fbgp-qpvv], here we provide a full account of our theoretical calculations and simulation results. We include derivations of optimal protocols, work variance, impact of measurement uncertainty, and information-acquisition costs.