Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Localized and degenerate controls for the incompressible Navier-Stokes system

Vahagn Nersesyan, Manuel Rissel

TL;DR

The paper establishes global approximate controllability for the 2D incompressible Navier–Stokes equations on the torus with a physically localized, degenerate forcing driven by a fixed, finite set of time-dependent controls. It develops a novel framework that treats the velocity problem via a vorticity reduction and leverages a transport-controllability analysis with a generating drift built from observable Fourier-mode families, combined with Coron’s return method and torus-flow translations to funnel information through the control region. The main result shows that four time-dependent scalar controls (extended by a scaling parameter $\sigma$) suffice to steer from any initial to any target state within any accuracy, with explicit, universal control profiles that depend only on the control region; the remaining controls are recovered from the primary four. This work advances Agrachev’s open problem by providing a concrete, localized finite-dimensional control construction and a rigorous reduction to a linear transport problem, offering a viable path toward localized force controllability in fluid dynamics.

Abstract

We consider the global approximate controllability of the two-dimensional incompressible Navier-Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier-Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.

Localized and degenerate controls for the incompressible Navier-Stokes system

TL;DR

The paper establishes global approximate controllability for the 2D incompressible Navier–Stokes equations on the torus with a physically localized, degenerate forcing driven by a fixed, finite set of time-dependent controls. It develops a novel framework that treats the velocity problem via a vorticity reduction and leverages a transport-controllability analysis with a generating drift built from observable Fourier-mode families, combined with Coron’s return method and torus-flow translations to funnel information through the control region. The main result shows that four time-dependent scalar controls (extended by a scaling parameter ) suffice to steer from any initial to any target state within any accuracy, with explicit, universal control profiles that depend only on the control region; the remaining controls are recovered from the primary four. This work advances Agrachev’s open problem by providing a concrete, localized finite-dimensional control construction and a rigorous reduction to a linear transport problem, offering a viable path toward localized force controllability in fluid dynamics.

Abstract

We consider the global approximate controllability of the two-dimensional incompressible Navier-Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier-Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.
Paper Structure (42 sections, 10 theorems, 165 equations, 7 figures)

This paper contains 42 sections, 10 theorems, 165 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The controllability problem
  3. Notations
  4. Main result
  5. Methodology
  6. Related literature
  7. Organization of this article
  8. Transport with generating drift: finite-dimensional controls
  9. Saturation property
  10. Vector fields obtained from observable families
  11. Construction of $\overline{\bm{y}}^{\star}$.
  12. Existence of finite-dimensional controls
  13. Step 1. The idea for showing $\operatorname{ker}(A_1^*) = \{0\}$.
  14. Step 2. Induction base.
  15. Step 3. Induction step.
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $r \in \mathbb{N}_0$, $T_{\operatorname{ctrl}} > 0$, $\bm{u}_0, \bm{u}_1 \in \mathbf{V}^{r}$, $\bm{f} \in {\rm L}^{2}((0,T_{\operatorname{ctrl}});{\rm H}^{r+2}(\mathbb{T}^2;\mathbb{R}^2))$, $\nu > 0$, and $\varepsilon > 0$. There exist such that the unique solution $\bm{u} \in {\rm C}^{0}([0,T_{\operatorname{ctrl}}];\mathbf{V}^{r})\cap{\rm L}^{2}((0,T_{\operatorname{ctrl}});\mathbf{V}^{r+1})$

Figures (7)

  • Figure 1: Two of the various possibilities for localizing the controls. In each picture, the (blue) filled part illustrates a valid control region $\Omega \subset \mathbb{T}^2$, which can be taken of arbitrary nonzero area.
  • Figure 2: An illustration of the dependencies in the proposed framework.
  • Figure 3: An illustration of the chosen covering of $\mathbb{T}^2$, here by $K = 36$ overlapping squares. Overlaps due to periodicity are filled with a corresponding pattern, and only a few squares are depicted. The (red) reference square $\mathcal{O}$ is located in the control region (indicated by blue dashes) and $\bm{p}_{\omegaup}$ marks its bottom left corner.
  • Figure 4: The vector field $\overline{\bm{y}}$, which is a function only of time, is constructed by a sequence of horizontal and vertical shifts of the whole torus $\mathbb{T}^2$.
  • Figure 5: A sketch of several ideas related to the identities \ref{['equation:ChangeOfVariables']} and \ref{['equation:UsingPropertyP3ToGet_chi']}, referring to \ref{['subsection:opencoveringoverlapping']} for the notations. In the particular example displayed here, the point $\bm{q} \in \mathcal{O}_i$ is located on the boundary of $\mathcal{O}_i\cap\mathcal{O}_{l}$ for some $l \in \{1,\dots,K\}\setminus\{i\}$. Thus, the integral over $[T_b,1]$ of $g(\bm{\mathcal{U}}(\bm{x},0,\cdot),\cdot)$ can be compressed into an integral over $(t^i_a, t^i_b)$. During this short interval, the square $\mathcal{O}_i$ has already been moved by $\bm{\mathcal{Y}}$ into $\mathcal{O}$ and the flow $\bm{\mathcal{Y}}$ pauses. The overlapping squares at the bottom left indicate all members of the family $(\mathcal{O}_l)_{l\in\{1,\dots,K\}}$ which intersect $\mathcal{O}_i$.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 17 more