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Approximate controllability on the group of volume-preserving diffeomorphisms

Andrei Agrachev, Bettina Kazandjian

TL;DR

This work addresses approximate controllability on the group of volume-preserving diffeomorphisms of the torus ${\mathbb T}^d$ for the affine system $\dot x=f(x)+u(t)$ with a fixed divergence-free drift $f$ and constant controls $u(t)$. By decomposing $f$ into finite Fourier modes and analyzing the closure of the associated Lie algebra, the authors identify invariant volume-preserving subgroups ${\mathrm Diff}_0{\mathbb T}^d_\Gamma$ and prove approximate controllability within these subgroups for $d=2,3$ when ${\mathcal M}_f$ generates ${\mathbb Z}^d$. They further show that for a residual set of $f$, the system is globally controllable for any finite ensemble of points in ${\mathbb T}^d$, although time bounds for such transfers cannot be universally guaranteed. The results hinge on translating the control problem into a linear-in-control ensemble framework, employing Rashevsky–Chow type arguments, and performing detailed Fourier-based Lie algebra computations in low dimensions, with explicit structure in $\mathfrak L_f$ described in both $d=2$ and $d=3$ cases. The work provides insight into controllability of turbulent-like flows via structured perturbations and lays groundwork for random perturbations in a follow-up study.

Abstract

We study controlability issues for the group of volume-preserving diffeomorphisms of the torus $\mathbb T^d$ for system $\dot x=f(x)+u(t)$, where $f$ is a fixed divergence free vector field on $\mathbb T^d$ and $u(t)$ are constant vector fields which generate translations of the torus. Main results concern $d$ equals two or three.

Approximate controllability on the group of volume-preserving diffeomorphisms

TL;DR

This work addresses approximate controllability on the group of volume-preserving diffeomorphisms of the torus for the affine system with a fixed divergence-free drift and constant controls . By decomposing into finite Fourier modes and analyzing the closure of the associated Lie algebra, the authors identify invariant volume-preserving subgroups and prove approximate controllability within these subgroups for when generates . They further show that for a residual set of , the system is globally controllable for any finite ensemble of points in , although time bounds for such transfers cannot be universally guaranteed. The results hinge on translating the control problem into a linear-in-control ensemble framework, employing Rashevsky–Chow type arguments, and performing detailed Fourier-based Lie algebra computations in low dimensions, with explicit structure in described in both and cases. The work provides insight into controllability of turbulent-like flows via structured perturbations and lays groundwork for random perturbations in a follow-up study.

Abstract

We study controlability issues for the group of volume-preserving diffeomorphisms of the torus for system , where is a fixed divergence free vector field on and are constant vector fields which generate translations of the torus. Main results concern equals two or three.
Paper Structure (12 sections, 18 theorems, 119 equations)

This paper contains 12 sections, 18 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Motion planning in the group of diffeomorphisms and controllability of finite ensembles of points
  3. Motion planning in the group of diffeomorphisms
  4. Controllability of finite ensembles of points
  5. Volume-preserving diffeomorphisms on ${\mathbb T}^d$
  6. Subgroups of volume-preserving flows on ${\mathbb T}^d$
  7. Approximation of volume-preserving diffeomorphisms by an affine in control system
  8. Proof of \ref{['theorem : affine system approx diffeo']}
  9. Proof of \ref{['theorem : affine system ensemble controllability']}
  10. Proof of \ref{['theorem : Lie algebra T2 and T3']}
  11. Bi-dimensional torus
  12. Tri-dimensional torus

Key Result

Theorem 1

Let $\mathcal{F}=\left\{F_1,\dots,F_s\right\}\subset\mathrm{Vec}M$. Let $t\mapsto V_t\in\overline{\mathrm{Lie}\mathcal{F}},\ t\in[0,T]$, be a time-varying vector field. Then for every $m\in {\mathbb N}$ and $\varepsilon>0$ there exists a control $t \mapsto u(t)=(u_1(t),\dots,u_s(t)) \in L^\infty([0,

Theorems & Definitions (39)

  • Remark 1
  • Theorem 1
  • proof
  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • ...and 29 more