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Approximate controllability for 2D Euler equations

Sérgio S. Rodrigues

TL;DR

This work proves that approximate controllability for the 2D Euler equations on bounded planar domains follows from the existence of a saturating subset of actuators under a relaxed definition. It adapts the Agrachev–Sarychev framework to domains with boundary by using a vorticity-based reduction and oblique projections, and provides a concrete saturating example for the unit disk. The results establish a pathway to finite-dimensional control of incompressible flows on bounded domains without requiring actuator eigenfunctions or special boundary operators. The paper also outlines extensions to Navier–Stokes and discusses domain-smoothness, norms, and actuator localization as directions for future work.

Abstract

Approximate controllability of the Euler equations is investigated by means of a finite set of actuators. It is proven that approximate controllability holds if we can find a saturating subset of actuators. The notion of saturating set is relaxed when compared to previous literature, still being a sufficient condition for approximate controllability. The result holds for general bounded two-dimensional spatial domains with smooth boundary. An example of a saturating set is given in the case the spatial domain is the unit disk.

Approximate controllability for 2D Euler equations

TL;DR

This work proves that approximate controllability for the 2D Euler equations on bounded planar domains follows from the existence of a saturating subset of actuators under a relaxed definition. It adapts the Agrachev–Sarychev framework to domains with boundary by using a vorticity-based reduction and oblique projections, and provides a concrete saturating example for the unit disk. The results establish a pathway to finite-dimensional control of incompressible flows on bounded domains without requiring actuator eigenfunctions or special boundary operators. The paper also outlines extensions to Navier–Stokes and discusses domain-smoothness, norms, and actuator localization as directions for future work.

Abstract

Approximate controllability of the Euler equations is investigated by means of a finite set of actuators. It is proven that approximate controllability holds if we can find a saturating subset of actuators. The notion of saturating set is relaxed when compared to previous literature, still being a sufficient condition for approximate controllability. The result holds for general bounded two-dimensional spatial domains with smooth boundary. An example of a saturating set is given in the case the spatial domain is the unit disk.
Paper Structure (46 sections, 19 theorems, 223 equations)

This paper contains 46 sections, 19 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Approximate controllability and saturating sets
  3. Novelties
  4. Euler versus Navier--Stokes equations
  5. Related literature
  6. Contents and notation
  7. Definitions and main results
  8. Proof of Theorem \ref{['T:approxcontrol']}
  9. Auxiliary results
  10. Vorticity function
  11. On existence and uniqueness of solutions
  12. The Stokes operator under Lions boundary conditions
  13. On the nonlinear term
  14. Particular basis for the subspaces ${\mathcal{G}}^j$
  15. On time-average of functions weighted with oscillating functions
  16. ...and 31 more sections

Key Result

Theorem 2.4

Let $G_0$ be a $(H\bigcap{\mathcal{C}}^{2,1}(\overline\Omega))$-saturating set as in Definition D:saturVlin-vort, such that Let also $T>0$ and $f\in {\mathcal{C}}^{\infty;1,1}([0,T]\times\overline\Omega)\bigcap L^\infty((0,T),H)$. Then, system evol-sys-U is approximately controllable at time $T$. Furthermore, we can take $u\in {\mathcal{C}}^\infty([0,T],{\mathbb R}^M)$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Corollary 3.4
  • proof
  • ...and 18 more