arXiv subjects

Lucas M. Moschen

Publications and source records attributed to Lucas M. Moschen.

5 recordsLinked to original sources

arXiv2026

Control Strategies for Multi-Species Wasserstein Gradient Flows

We stabilize stationary states of coupled multi-species McKean--Vlasov equations by feedback control, extending a spectral approach based on the Wasserstein Hessian and a Riccati equation from a single density to a product of Wasserstein spaces. We give necessary and sufficient conditions for local exponential stabilization at a prescribed rate when only one species is actuated, together with the minimum number of control inputs required. For two species, the condition reduces to a rank condition on the Fourier coefficients of the cross-interaction kernel over the eigenspaces of the unactuated species' linearized dynamics. A two-community noisy Kuramoto model and a cell-sorting model illustrate the results.

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arXiv2026

Feedback Control and Local Convexification of Wasserstein Gradient Flows

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.

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arXiv2025

Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs

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.

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arXiv2025

A Spectral Approach to Optimal Control of the Fokker-Planck Equation

In this paper, we present a spectral optimal control framework for Fokker-Planck equations based on the standard ground state transformation that maps the Fokker-Planck operator to a Schrodinger operator. Our primary objective is to accelerate convergence toward the (unique) steady state. To fulfill this objective, a gradient-based iterative algorithm with Pontryagin's maximum principle and the Barzilai-Borwein update is developed to compute time-dependent controls. Numerical experiments on two-dimensional ill-conditioned normal distributions and double-well potentials demonstrate that our approach effectively targets slow-decaying modes, thus increasing the spectral gap.

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arXiv2020

A model for COVID-19 with isolation, quarantine and testing as control measures

In this article we propose a compartmental model for the dynamics of Coronavirus Disease 2019 (COVID-19). We take into account the presence of asymptomatic infections and the main policies that have been adopted so far to contain the epidemic: isolation (or social distancing) of a portion of the population, quarantine for confirmed cases and testing. We model isolation by separating the population in two groups: one composed by key-workers that keep working during the pandemic and have a usual contact rate, and a second group consisting of people that are enforced/recommended to stay at home. We refer to quarantine as strict isolation, and it is applied to confirmed infected cases. In the proposed model, the proportion of people in isolation, the level of contact reduction and the testing rate are control parameters that can vary in time, representing policies that evolve in different stages. We obtain an explicit expression for the basic reproduction number $\mathcal{R}_0$ in terms of the parameters of the disease and of the control policies. In this way we can quantify the effect that isolation and testing have in the evolution of the epidemic. We present a series of simulations to illustrate different realistic scenarios. From the expression of $\mathcal{R}_0$ and the simulations we conclude that isolation (social distancing) and testing among asymptomatic cases are fundamental actions to control the epidemic, {and the stricter these measures are and the sooner they are implemented,} the more lives can be saved. Additionally, we show that people that remain in isolation significantly reduce their probability of contagion, so risk groups should be recommended to maintain a low contact rate during the course of the epidemic.

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