コロキウム

The Kuramoto model (KM) defined on graphs presents mathematical models in various fields such as physics, biology, engineering, and social sciences. Since its dimensions are typically very high in actual applications, the continuum limit, which is a single integro-differential equation and provides approximate solutions to the KM, is often useful. In this talk we study feedback control of the Kuramoto model (KM) with natural frequencies on a uniform graph that 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). For the complete simple graph, we prove that if the feedback gain is larger than a critical value, then the KM has an asymptotically stable synchronized solution that tends to the target orbit as the feedback gain goes to infinity, and that the CL has the corresponding asymptotically stable continuous solution. Moreover, for random dense and sparse graphs, we show that the continuous solution to the same CL as for the complete simple graph behaves like an asymptotically stable one in the KM. We demonstrate the theoretical results by numerical simulations for the KM on the three types of graphs.