  Based on analytical and numerical calculations we study the dynamics of an
overdamped colloidal particle moving in two dimensions under time-delayed,
non-linear feedback control. Specifically, the particle is subject to a force
derived from a repulsive Gaussian potential depending on the difference between
its instantaneous position, $\mathbf{r}(t)$, and its earlier position
$\mathbf{r}(t-\tau)$, where $\tau$ is the delay time. Considering first the
deterministic case, we provide analytical results for both, the case of small
displacements and the dynamics at long times. In particular, at appropriate
values of the feedback parameters, the particle approaches a steady state with
a constant, non-zero velocity whose direction is constant as well. In the
presence of noise, the direction of motion becomes randomized at long times,
but the (numerically obtained) velocity autocorrelation still reveals some
persistence of motion. Moreover, the mean-squared displacement (MSD) reveals a
mixed regime at intermediate times with contributions of both, ballistic motion
and diffusive translational motion, allowing us to extract an estimate for the
effective propulsion velocity in presence of noise. We then analyze the data in
terms of exact, known results for the MSD of active Brownian particles. The
comparison indeed indicates a strong similarity between the dynamics of the
particle under repulsive delayed feedback and active motion. This relation
carries over to the behavior of the long-time diffusion coefficient
$D_\mathrm{eff}$ which, similarly to active motion, is strongly enhanced
compared to the free case. Finally we show that, for small delays,
$D_\mathrm{eff}$ can be estimated analytically.
