We study the feedback stabilization of the McKean-Vlasov PDE on the torus. Our goal is to steer the dynamics toward a prescribed stationary distribution or accelerate convergence to it using a time-dependent control potential. We reformulate the controlled PDE in a weighted-projected space and apply the ground-state transform to obtain a Schrodinger-type operator. The resulting operator framework enables spectral analysis, verification of the infinite-dimensional Hautus test, and the construction of Riccati-based feedback laws. We rigorously prove local exponential stabilization via maximal regularity arguments and nonlinear estimates. Numerical experiments on well-studied models (the noisy Kuramoto model for synchronization, the O(2) spin model in a magnetic field, and the Gaussian/von Mises attractive interaction potential) showcase the effectiveness of our control strategy, demonstrating convergence speed-ups and stabilization of otherwise unstable equilibria.

翻译：本文研究环面上McKean-Vlasov偏微分方程的反馈镇定问题。我们的目标是通过时变控制势函数，将系统动力学导向预设的稳态分布或加速其收敛过程。我们将受控偏微分方程重构于加权投影空间，并应用基态变换得到Schrodinger型算子。该算子框架支持谱分析、无限维Hautus检验验证以及基于Riccati方程的反馈律构造。通过极大正则性论证和非线性估计，我们严格证明了局部指数镇定性。在经典模型（同步噪声Kuramoto模型、磁场中的O(2)自旋模型、高斯/von Mises吸引相互作用势）上的数值实验验证了控制策略的有效性，展示了收敛加速效应及对原不稳定平衡点的镇定能力。