Feedback control of the Kuramoto model with uniformly spaced natural frequencies defined on uniform graphs

We study feedback control of the Kuramoto model (KM) with natural
frequencies on a uniform graph which may be complete simple, random
dense or random sparse. We choose as the target orbit the synchronized
state in which all oscillators rotate with the same rotational speed,
and design the controller using the continuum limit (CL).  When the
graph is complete simple, we prove that if the feedback gain is larger
than a critical value, then there exists an asymptotically stable
synchronized solution that tends to the target orbit as the feedback
gain goes to infinity, and that the CL has an asymptotically stable
continuous solution which corresponds to the asymptotically stable
solution to the KM. When the graph is random, we show that the
continuous solution to the same CL as in the above case behaves as an
asymptotically stable one in the KM.  We demonstrate the theoretical
results by numerical simulations for the KM on the three types of
graphs.