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Stabilization via localized controls in nonlocal models of crowd dynamics

Nikolay Pogodaev, Francesco Rossi

Abstract

We consider a control system driven by a nonlocal continuity equation. Admissible controls are Lipschitz vector fields acting inside a fixed open set. We demonstrate that small perturbations of the initial measure, traced along Wasserstein geodesics, may be neutralized by admissible controls. More specifically, initial perturbations of order $\varepsilon$ can be reduced to order $\varepsilon^{1+κ}$, where $κ$ is a positive constant.

Stabilization via localized controls in nonlocal models of crowd dynamics

Abstract

We consider a control system driven by a nonlocal continuity equation. Admissible controls are Lipschitz vector fields acting inside a fixed open set. We demonstrate that small perturbations of the initial measure, traced along Wasserstein geodesics, may be neutralized by admissible controls. More specifically, initial perturbations of order can be reduced to order , where is a positive constant.
Paper Structure (6 sections, 9 theorems, 29 equations, 2 figures)

This paper contains 6 sections, 9 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. Wasserstein space and its geodesics
  4. Basic definitions and properties
  5. Prolongation of geodesics
  6. Main result

Key Result

Theorem I.2

Let a nonlocal vector field $V$, an open set $\omega\subset \mathbb{R}^d$ and an absolutely continuous measure $\varrho_0\in \mathcal{P}_c(\mathbb{R}^d)$ satisfy Assumptions $(A_{1,2})$. Let $\Pi(\varrho_0)$ be defined as in eq:Pi for some convex compact set $\Omega \subset \mathbb{R}^d$. Then, ther

Figures (2)

  • Figure 1: The concept of stabilization: we perturb $\varrho_0$ to a new measure $\varrho$. We reduce the perturbation rate with a Lipschitz control localized in $\omega$.
  • Figure 2: Geometric Condition $(A_2)$: any trajectory of $V_t(\mu_t)$ issuing from $\mathop{\rm spt}(\varrho_0)$ crosses $\omega$ by the time $T$.

Theorems & Definitions (28)

  • Definition I.1
  • Theorem I.2
  • Theorem I.3: PR2024
  • Definition II.1
  • Definition II.2
  • Definition II.3
  • Definition II.4
  • Theorem II.5: villaniTopicsOptimalTransportation2003
  • Definition II.6
  • Definition II.7
  • ...and 18 more