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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.13588 |
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| _version_ | 1866915862059941888 |
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| author | Kalise, Dante Moschen, Lucas M. Pavliotis, Grigorios A. |
| author_facet | Kalise, Dante Moschen, Lucas M. Pavliotis, Grigorios A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13588 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Feedback Control and Local Convexification of Wasserstein Gradient Flows Kalise, Dante Moschen, Lucas M. Pavliotis, Grigorios A. Optimization and Control Analysis of PDEs Dynamical Systems Functional Analysis Primary: 35Q84, 49J20, 93D15. Secondary: 35B40, 49Q22 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. |
| title | Feedback Control and Local Convexification of Wasserstein Gradient Flows |
| topic | Optimization and Control Analysis of PDEs Dynamical Systems Functional Analysis Primary: 35Q84, 49J20, 93D15. Secondary: 35B40, 49Q22 |
| url | https://arxiv.org/abs/2603.13588 |