Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

TL;DR

This paper introduces a spectral analysis-based feedback control method for stabilizing McKean-Vlasov PDEs, enabling convergence acceleration and stabilization of unstable equilibria through Riccati-based laws.

Contribution

It develops a novel control framework using spectral analysis and Riccati equations for local exponential stabilization of McKean-Vlasov PDEs.

Findings

Successfully stabilized models in 1D and 2D, including Kuramoto and spin models.

Achieved convergence acceleration and stabilization of unstable equilibria.

Validated the approach with numerical experiments demonstrating effectiveness.

Abstract

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.