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arXiv · 2507.12411

Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

Dante Kalise·Lucas M. Moschen·Grigorios A. Pavliotis
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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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Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis. 2025-07-16. Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs. https://arxiv.org/abs/2507.12411

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