Trajectory stabilization of nonlocal continuity equations by localized controls

2024/03/05 by Nikolay Pogodaev, Pogodaev, Nikolay, Francesco Rossi +1
Computer Science · Engineering · Mathematics · #35Q9 #93C20 #93D20 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2403.02837

openalex publication_date 2024/03/05 · openalex created_date 2024/03/07 · openalex updated_date 2026/07/28

Abstract

We discuss stabilization around trajectories of the continuity equation with nonlocal vector fields, where the control is localized, i.e., it acts on a fixed subset of the configuration space. We first show that the correct definition of stabilization is the following: given an initial error of order ε, measured in Wasserstein distance, one can improve the final error to an order ε^1+κ with κ>0. We then prove the main result: assuming that the trajectory crosses the subset of control action, stabilization can be achieved. The key problem lies in regularity issues: the reference trajectory needs to be absolutely continuous, while the initial state to be stabilized needs to be realized by a small Lipschitz perturbation or being in a very small neighborhood of it.

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