Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

Abstract: 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.