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Controllability and kinetic limit of spherical particles immersed in a viscous fluid

Marta Zoppello, Henry Shum, Marco Morandotti

Abstract

This paper deals with systems of spherical particles immersed in a viscous fluid. Two aspects are studied, namely the controllability of such systems, with particular attention to the case of one active particle and either one or two passive ones, and the kinetic limit of such systems as the number of particles diverges. The former issue is tackled in the framework of geometric control theory, whereas the latter resorts to Boltzmann-type formulations of the system of interacting particles.

Controllability and kinetic limit of spherical particles immersed in a viscous fluid

Abstract

This paper deals with systems of spherical particles immersed in a viscous fluid. Two aspects are studied, namely the controllability of such systems, with particular attention to the case of one active particle and either one or two passive ones, and the kinetic limit of such systems as the number of particles diverges. The former issue is tackled in the framework of geometric control theory, whereas the latter resorts to Boltzmann-type formulations of the system of interacting particles.

Paper Structure

This paper contains 10 sections, 5 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The model
  4. Two controllability results
  5. One active and one passive particle
  6. One active and two passive particles
  7. Kinetic limit
  8. The optimal control problem: binary control and Boltzmann approach
  9. The binary optimal control
  10. The Boltzmann approach for the infinite dimensional optimal control

Key Result

Theorem 2.2

Let $\Omega\subseteq \mathbb{R}^n$ be an open connected set. Assume that the vector fields $\mathbf{g}^1,\ldots,\mathbf{g}^m\in C^\infty(\Omega,\mathbb{R}^n)$ generate a Lie algebra $\mathfrak{Lie}\{\mathbf{g}^1,\ldots,\mathbf{g}^m\}$ that satisfies the Lie algebra rank condition Then system 001 is controllable in $\Omega$.

Theorems & Definitions (13)

  • Definition 2.1: Coron
  • Theorem 2.2: Chow-Rashewskii, Agrachev
  • Definition 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Definition 5.1
  • Theorem 5.2
  • proof
  • ...and 3 more