Feedback Control and Local Convexification of Wasserstein Gradient Flows

TL;DR

This paper develops a spectral and control-theoretic framework for stabilizing Wasserstein gradient flows of free energies on the flat torus, enabling exponential convergence to stationary states through feedback control.

Contribution

It introduces a spectral analysis of the Wasserstein Hessian and designs a feedback law via Riccati equations to ensure local exponential stability of the flow.

Findings

Wasserstein Hessian at stationary states is a self-adjoint operator with compact resolvent.

A finite-rank feedback law can shift the Hessian spectrum above any threshold.

The closed-loop flow converges locally exponentially to the stationary state.

Abstract

For free energies of the form \[ F(\mu) = E(\mu) + \sigma\int_\Omega \mu\log\mu\,dx, \quad \sigma > 0, \] we study the Wasserstein gradient flow, a continuity equation also known as mean-field Langevin dynamics, around a stationary state $\bar\mu$ on the flat torus. Our first result identifies the Wasserstein Hessian of $F$ at $\bar\mu$ with a self-adjoint operator with compact resolvent on a Hilbert space of potential variables, and shows that, up to the natural Riesz isometry, this operator generates the linearized gradient flow. This spectral description allows us to design a finite-rank feedback law, via an algebraic Riccati equation, that shifts the closed-loop Hessian spectrum above any prescribed threshold $\delta > 0$. As a consequence, the nonlinear closed-loop flow converges locally exponentially to $\bar\mu$ with rate $\delta$. Under an additional second-order remainder assumption on the first variation, the corresponding closed-loop energy is also locally strongly convex in chart coordinates. We illustrate the framework on the flat torus and discuss extensions to multi-species systems, moment-constrained Fokker-Planck equations, and closed Riemannian manifolds.