Feedback Control and Local Convexification of Wasserstein Gradient Flows

Abstract

Building on the feedback stabilization framework that we developed for McKean-Vlasov PDEs, we treat a class of entropy-regularized free energies $F = E + \sigma\mathrm{Ent}$ on the flat torus, with $E$ possibly nonconvex and nonlocal, and show that the resulting controlled system admits a gradient flow formulation in Wasserstein space. Specifically, we realize the Wasserstein Hessian at a target stationary measure $\bar\mu$ as a self-adjoint operator with compact resolvent and show that its negative is unitarily equivalent to the generator of the linearized dynamics. The resulting feedback stabilization can then be interpreted as a local convexification of the free energy landscape: for any prescribed threshold $\delta > 0$, the control, obtained from an algebraic Riccati equation for the linearized problem, induces a finite-rank perturbation of the Wasserstein Hessian that lifts its spectrum above $\delta$, yielding local displacement convexity on a H\"older neighborhood of $\bar\mu$ in regimes where the uncontrolled energy is nonconvex or slowly contracting. Under a local well-posedness assumption, $\bar\mu$ is moreover a locally exponentially stable equilibrium of the closed-loop dynamics, with rate at least $\delta$. The framework covers McKean-Vlasov models and extends to moment-constrained Fokker-Planck dynamics and Fokker-Planck equations on closed Riemannian manifolds.