Optimal work control for an activity-driven odd tracer in a viscoelastic bath

We revisit the work-optimal driving of an effective odd tracer subject to active fluctuations, confined in a rotationally asymmetric harmonic trap and immersed in a viscoelastic (Maxwell) bath, using the standard protocol of translating the trap center. The odd mobility of an odd tracer need not be an intrinsic single-particle property, but can instead emerge from coarse graining over particles coupled to a viscoelastic bath. Two control schemes are considered: open loop (without measuring the active force) and closed loop (conditioned on the initial value of the active force). In the open-loop case, we show that viscoelastic memory increases the energetic cost and breaks time-reversal symmetry in the individual components of the optimal protocol, while the sum of the two protocol components can remain time symmetric. In closed-loop control, an initial measurement of the active force determines the strategy adopted by the optimal protocol. We identify the regimes of force configuration, the timescales associated with the persistence of the active force and bath relaxation, and the tracer’s odd mobility that govern optimal performance—maximizing work extraction or, equivalently, minimizing the control input. In particular, for odd tracers with specific orientations of the initial force, the interplay between the persistence time and the bath relaxation time becomes crucial, selecting an intermediate persistence that optimizes the work cost. In strongly viscoelastic regimes, we find that the cost generally increases with the bath relaxation time, reflecting the energetic penalty associated with long memory. Our results offer practical design principles for optimizing active micromachines in complex biomolecular environments, illustrating how measurement, memory, and controlled tuning of persistence can be harnessed for efficient energy conversion.