Feedback Control and Local Convexification of Wasserstein Gradient Flows

Abstract

For free energies of the form \[ F(μ) = E(μ) + σ\int_Ωμ\logμ\,dx, \quad σ> 0, \] we study the Wasserstein gradient flow, a continuity equation also known as mean-field Langevin dynamics, around a stationary state $\barμ$ on the flat torus. Our first result identifies the Wasserstein Hessian of $F$ at $\barμ$ 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 $δ> 0$. As a consequence, the nonlinear closed-loop flow converges locally exponentially to $\barμ$ with rate $δ$. 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.

Figures (8)

• Figure 1: Control-induced convexification of the free energy. Slice of the free energy near $\bar{\mu}$ along the first eigendirection (quartic double-well on the torus). The Riccati feedback acts on the dynamics, modifying the gradient flow via a finite-rank force field. As a consequence, the linearized feedback-controlled dynamics coincide with the gradient flow of a feedback-modified energy (orange), whose Hessian at $\bar{\mu}$ is strictly positive.
• Figure 2: Feedback-induced convexification of the Kuramoto model $\partial_t \mu = \sigma \Delta \mu + K \nabla \cdot (\mu \nabla (W * \mu))$ with $W(x) = -\cos(x)$. We plot the restricted free energy $\mathcal{E}$ along the geodesic slice induced by the principal unstable eigenfunction $\phi_1$. For $K=3$ and $\sigma=0.5$, the uniform incoherent state $\bar{\mu} = 1/2\pi$ is an unstable equilibrium ($\lambda_1 < 0$), corresponding to a local maximum of the uncontrolled energy (blue). The closed-loop energy (red) adds a quadratic penalty $\frac{1}{2}\langle \Pi P\xi, \xi\rangle_X$ that restores strict local convexity at the equilibrium.
• Figure 3: Multi-species coupled dynamics. Two species on $\mathbb{T}$ with $V(x) = -2\cos(4\pi x)$, $\sigma = 0.5$, cross-interaction $W_{ij}(x) = -0.5\cos(2\pi x)$. Left: joint product-space spectrum of the uncontrolled operator $A_{\mathrm{ms}}$ and the rank-2 closed-loop operator $A_{\mathrm{ms}}+\Pi_{\mathrm{ms}}$. Right: joint perturbation norm $\|(\xi^1_N(t), \xi^2_N(t))\|_{X\times X} = \bigl(\|\xi^1_N(t)\|_X^2 + \|\xi^2_N(t)\|_X^2\bigr)^{1/2}$ for the uncontrolled and controlled coupled systems. For the target rate $\delta = 5$, the two-control feedback lifts the product-space spectral gap from approximately $0.036$ to $\lambda_\Pi \approx 5.01 > \delta$.
• Figure 4: Feedback stabilization on the sphere $\mathbb{S}^2$ for the non-convex potential $V(\theta) = -\alpha \cos^2\theta$. We plot the $X$-norm of the density perturbation for the uncontrolled (black, $\lambda_1 \approx 0.13$) and controlled (red, $\lambda_{\Pi} \approx 2.10$) dynamics with $\alpha=2$ and $\sigma=0.5$. The negative curvature near the equator ($\mathrm{Ric}_{V,\sigma} < 0$) leads to metastable behavior in the uncontrolled system. For the target rate $\delta = 2$, the rank-1 feedback yields a closed-loop gap $\lambda_\Pi > \delta$, significantly accelerating the relaxation to the polar-concentrated equilibrium $\bar{\mu}$.
• Figure 5: Spectral shift and cost analysis for the double-well potential $V(x)=-2\cos(4\pi x)$ on $\mathbb{T}$ with $\sigma=0.5$. Discretization: $N_{\text{fine}}=512$, $N=10$ Galerkin modes (ETD1). In this sweep we use the one-mode Riccati family $Q\phi_1 = m_1\phi_1$ with $m_1=\delta^2$. Left: the observed decay rate $\lambda_{\mathrm{obs}}$ increases with $\delta$ and matches the theoretical prediction $\lambda_{\Pi} = \delta + \sqrt{(\lambda_1-\delta)^2 + m_1}$. Center: the control cost $\mathcal{J}_u = \frac{1}{2}\int_0^T |u|^2\,dt$ grows with $\delta$. Right: the settling time $t_{99\%}$ decreases monotonically with the feedback gain.

Theorems & Definitions (79)

• Theorem A: Hessian realization and spectral structure
• Theorem B: Linearized flow in density variables
• Theorem C: Spectral shift and closed-loop Hessian coercivity
• Theorem D: Local exponential stabilization
• Theorem E: Chart-level strong convexity under Assumption \ref{['ass:E5']}
• Lemma 2.1: Existence of stationary points
• proof
• Example 2.2
• Lemma 2.3: Regularity and bounds of $\bar{\mu}$
• proof