| Abstract: |
| In this presentation, we consider a mean-field optimal control problem for a consensus dynamics of high-dimensional Kuramoto-type with diffusion on the unit sphere. The control acts through a prescribed drift field and an interaction gain, and the cost functional is given to track a given target density while penalizing the control effort. At the microscopic level, we formulate the corresponding controlled $N$-particle Liouville problem and establish the existence of optimal controls. For fixed controls, we obtain a quantitative stochastic mean-field limit showing that the one-particle marginal converges to the solution of the mean-field equation with the convergence rate $\mathcal O(1/\sqrt{N})$. Finally, we show that microscopic optimal controls approximate a mean-field optimal control: any weak limit of particle-level minimizers is optimal for the mean-field problem. |
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