Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

2025/07/16 by Dante Kalise, Kalise, Dante, Lucas M. Moschen +3
Engineering · #35K55 #35Q84 #49N10 #93C20 #93D15 #Advanced Control Systems Optimization #FOS: Mathematics #FOS: Physical sciences #G.1.6 #G.1.8 #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2507.12411

openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/08/01

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.

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• Fredholm Transform and Local Rapid Stabilization for a Kuramoto-Sivashinsky Equation
• On the stability of the invariant probability measures of McKean-Vlasov equations