  Closed-loop or feedback control ratchets use information about the state of
the system to operate with the aim of maximizing the performance of the system.
In this paper we investigate the effects of a time delay in the feedback for a
protocol that performs an instantaneous maximization of the center-of-mass
velocity. For the one and the few particle cases the flux decreases with
increasing delay, as an effect of the decorrelation of the present state of the
system with the information that the controller uses, but the delayed
closed-loop protocol succeeds to perform better than its open-loop counterpart
provided the delays are smaller than the characteristic times of the Brownian
ratchet. For the many particle case, we also show that for small delays the
center-of-mass velocity decreases for increasing delays. However, for large
delays we find the surprising result that the presence of the delay can improve
the performance of the nondelayed feedback ratchet and the flux can attain the
maximum value obtained with the optimal periodic protocol. This phenomenon is
the result of the emergence of a dynamical regime where the presence of the
delayed feedback stabilizes one quasiperiodic solution or several
(multistability), which resemble the solutions obtained in the so-called
threshold protocol. Our analytical and numerical results point towards the
feasibility of an experimental implementation of a feedback controlled ratchet
that performs equal or better than its optimal open-loop version.
