  We explore theoretically the navigation of an active particle based on
delayed feedback control. The delayed feedback enters in our expression for the
particle orientation which, for an active particle, determines (up to noise)
the direction of motion in the next time step. Here we estimate the orientation
by comparing the delayed position of the particle with the actual one. This
method does not require any real-time monitoring of the particle orientation
and may thus be relevant also for controlling sub-micron sized particles, where
the imaging process is not easily feasible. We apply the delayed feedback
strategy to two experimentally relevant situations, namely, optical trapping
and photon nudging. To investigate the performance of our strategy, we
calculate the mean arrival time analytically (exploiting a small-delay
approximation) and by simulations.
