Optimal Control for Kuramoto Model: from Many-Particle Liouville Equation to Diffusive Mean-Field Problem
Li Chen, Yucheng Wang, Valeriia Zhidkova
TL;DR
This work studies the mean-field optimal control of Kuramoto oscillator ensembles, connecting the stochastic particle dynamics to a diffusive nonlocal PDE on $S^1$ through wrapped distributions. By formulating both a microscopic Liouville control problem and a macroscopic mean-field control problem, the authors establish existence of optimizers at the particle level and prove a rigorous mean-field limit via $\Gamma$-convergence, supported by strong marginal convergence obtained from relative entropy and large-deviation estimates. They show that the first marginal converges to the mean-field density with an $O(1/N)$ rate, and that minimizers of the microscopic cost converge to minimizers of the mean-field cost, ensuring consistency across scales. The results provide a solid theoretical bridge linking multi-particle SDEs, Liouville equations, and nonlocal mean-field PDEs for controlled synchronization, with implications for designing controls that steer density toward target distributions on $S^1$.
Abstract
In this paper, we investigate the mean-field optimal control problem of a swarm of Kuramoto oscillators. Using the notion of wrapped distribution, we explain the connection between the stochastic particle system and the mean-field PDE on the periodic domain. In the limit of an infinite number of oscillators the collective dynamics of the agents' density is described by a diffusive mean-field model in the form of a non-local PDE, where the non-locality arises from the synchronization mechanism. We prove the existence of the optimal control of the mean-field model by using $Γ$-convergence strategy of the cost functional corresponding to the Liouville equation on the particle level. In the discussion of propagation of chaos for fixed control functions we complete the relative entropy estimate by using large deviation estimate given by \cite{MR3858403}.