Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs
For researchers in control theory and statistical physics, this provides a rigorous method to stabilize nonlinear PDEs arising from mean-field interactions, though the approach is incremental as it extends linearization-based techniques to a new class of PDEs.
This paper develops a feedback control framework for stabilizing McKean-Vlasov PDEs on the torus, achieving local exponential stabilization with a chosen convergence rate. Numerical experiments on models like the noisy Kuramoto model demonstrate convergence acceleration and stabilization of unstable equilibria.
We develop a feedback control framework for stabilizing 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, zero-mean 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 construction of a Riccati-based feedback law derived from the linearized dynamics, yielding local exponential stabilization with a chosen convergence rate. We rigorously prove local exponential stabilization via maximal regularity arguments and nonlinear estimates. Numerical experiments on well-studied models in one and two dimensions (the noisy Kuramoto model for synchronization, the O(2) spin model in a magnetic field, and the von Mises attractive interaction potential) showcase the effectiveness of our control strategy, demonstrating convergence acceleration and stabilization of unstable equilibria.