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Trajectory stabilization of nonlocal continuity equations by localized controls

Nikolay Pogodaev, Francesco Rossi

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 $\varepsilon$, measured in Wasserstein distance, one can improve the final error to an order $\varepsilon^{1+κ}$ 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.

Trajectory stabilization of nonlocal continuity equations by localized controls

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 , measured in Wasserstein distance, one can improve the final error to an order with . 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.
Paper Structure (17 sections, 15 theorems, 122 equations, 4 figures)

This paper contains 17 sections, 15 theorems, 122 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Ordinary differential equations and bi-Lipschitz homeomorphisms
  4. Notation
  5. Matrix ordinary differential equations and differential inequalities
  6. Bi-Lipschitz homeomorphisms
  7. Classical flows and regular perturbations
  8. Nonlocal flows
  9. Wasserstein space
  10. Proof of the main theorem
  11. Cut-off function
  12. Admissible control: linear case
  13. Admissible control: nonlocal case
  14. Stabilization property
  15. Eliminating Assumption $(A_{3})$
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let a nonlocal vector field $V$, an open set $\omega\subset \mathbb{R}^d$ and a measure $\varrho_0\in \mathcal{P}_c(\mathbb{R}^d)$ satisfy Assumptions $(A_{1,2})$. Then, there exist $\kappa_{*},\varepsilon_{*}>0$ such that the set $\Gamma_{\varepsilon_{*}}(\varrho_0)$ is $\kappa$-stabilizable around

Figures (4)

  • Figure 1: The concept of stabilization: we perturb $\varrho_0$ by a Lipschitz control $v$. Our goal is to neutralize this perturbation with another Lipschitz control acting inside $\omega$ only.
  • 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 $T$.
  • Figure 3: The set $\Gamma_{\varepsilon}(\varrho_0)$ (black curves) and its enlargement ${\bf B}_{r(\varepsilon,\kappa)}\left(\Gamma_{\varepsilon}(\varrho_0)\right)$ (grey region) inside ${\bf B}_{\varepsilon}(\varrho_0)$.
  • Figure 4: Optimal transport between $\varrho_{\varepsilon_3}$ and $\varrho_{\varepsilon_2}$, then between $\varrho_{\varepsilon_2}$ and $\varrho_{\varepsilon_1}$. Here $m_n=1/2^{n-1}$ is the mass of each atom of $\varrho_{\varepsilon_n}$, $d_n=1/2^n$ is the distance that each atom passes during the optimal mass transfer between $\varrho_{\varepsilon_n}$ and $\varrho_{\varepsilon_{n-1}}$.

Theorems & Definitions (39)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1: Generalized Grönwall inequality
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 29 more