  We present a linear model, which mimics the response of a spatially extended
dissipative medium to a distant perturbation, and investigate its dynamics
under delayed feedback control. The time a perturbation needs to propagate to a
measurement point is captured by an inherent delay time (or latency). A
detailed linear stability analysis demonstrates that a non-zero system delay
acts destabilizing on the otherwise stable fixed point for sufficiently large
feedback strengths. The imaginary part of the dominant eigenvalue is bounded by
twice the feedback strength. In the relevant parameter space it changes
discontinuously along specific lines when switching between branches of
eigenvalues. When the feedback control force is bounded by a sigmoid function,
a supercritical Hopf bifurcation occurs at the stability-instability
transition. The frequency and amplitude of the resulting limit cycles respond
to parameter changes like the dominant eigenvalue. In particular, they show
similar discontinuities along specific lines. These results are largely
independent of the exact shape of the sigmoid function. Our findings match well
with previously reported results on a feedback-induced instability of vortex
diffusion in a rotationally driven Newtonian fluid [M. Zeitz, P. Gurevich, and
H. Stark, Eur. Phys. J. E 38, 22 (2015)]. Thus, our model captures the
essential features of nonlocal delayed feedback control in spatially extended
dissipative systems.
