Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

TL;DR

The paper addresses stabilizing the overdamped McKean–Vlasov PDE on the torus toward a prescribed stationary distribution using a time-dependent control potential. It linearizes the dynamics around a steady state, maps the linearized generator to a Schrödinger-type operator via a ground-state transform, and verifies an infinite-dimensional Hautus test to justify a Riccati-based feedback u = −𝓑^*Π y that stabilizes the linearized system. Existence of a unique ARE solution Π and the resulting feedback yield local exponential stabilization of the nonlinear PDE, with a maximal-regularity framework used to control nonlinear terms. Numerical experiments in 1D and 2D (including noisy Kuramoto, symmetry-breaking perturbations, and Von Mises interactions) validate rapid convergence and stabilization, demonstrating practical efficacy and robustness of the approach.

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.

Figures (8)

• Figure 1: Time-evolution to the unstable steady state. Uncontrolled (left) vs. controlled (right) evolution of $\mu(x,t)$ for the noisy Kuramoto model with $\sigma=0.5$, $V=0$, $W=-K\cos( x)$. Black: initial density; gray: intermediate snapshots; red/blue: final density (uncontrolled/controlled).
• Figure 1: Different steady states and spectral gap. (a) Shape of the synchronized equilibrium densities $\bar{\mu}(\theta)$ as coupling $K$ increases. (b) Numerical estimates and theoretical lower bounds of the spectral gap $\lambda_{\rm gap}$ as a function of $K$.
• Figure 2: Results for the control for Noisy Kuramoto Model. (a) Acceleration to the incoherent state ($K=0.95$): Riccati feedback (blue) vs. uncontrolled (red dashed). (b) Stabilization of the unstable uniform state ($K=5$): feedback recovers exponential decay. (c) Steering under a different stable steady state (different translation) ($K=5$): feedback still maintains decay. All panels plot $\|y(t)\|_{\bar{\mu}^{-1}}$ on a log-scale.
• Figure 3: Results for the controlled noisy Kuramoto model with different couplings. Log-scale evolution of the scaled norm $\|y(t)\|_{\bar{\mu}^{-1}} / \|y(0)\|_{\bar{\mu}^{-1}}$ for three scenarios: (a) feedback-controlled convergence to the uniform steady-state, (b) feedback-controlled acceleration to the stable steady state, and (c) open-loop (uncontrolled) convergence to the same target.
• Figure 4: Spectral shift by the feedback control in the noisy Kuramoto model. Spectra with real part at least $-30$ of the uncontrolled generator $\mathcal{L}$ (red crosses) and the closed-loop operator $\mathcal{L} - \mathcal{B} \mathcal{B}^\ast \Pi$ (blue circles) for the noisy Kuramoto model with three coupling strengths $K$. The red dashed line marks $x=0$ (stability boundary) and the blue solid line marks $x=-\delta$ with $\delta=1$ (target decay rate).

Theorems & Definitions (30)

• Definition 2.2: Weak solution
• Theorem 2.3: Global well-posedness
• Proof 1
• Remark 2.4: Classical regularity under smooth data
• Proposition 2.5: Existence and uniqueness of stationary states
• Remark 2.6: Non-convex case and bifurcations
• Lemma 2.7: Spectral properties of $\mathcal{L}_{\mathrm{loc}}$
• Lemma 2.8
• Remark 2.9
• Proposition 2.10: Discrete spectrum of $\mathcal{H}$