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Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

Manuel Rissel, Ya-Guang Wang

TL;DR

This work studies global exact boundary controllability for ideal incompressible magnetohydrodynamic flows in a planar rectangle, with boundary controls active on the two vertical walls. By reformulating the MHD system in Elsässer variables and applying Coron's return method, the authors prove global exact controllability for the Elsässer system and then transfer the result to the original MHD system through a pressure-like corrector, addressing the challenging nonlinear coupling. A fixed-point scheme in a weighted, extended-domain framework, together with carefully constructed initial-data extensions, establishes a local null controllability result that is then scaled and glued to obtain the global, small-time controllability. The approach provides a first rigorous step toward boundary controllability of ideal MHD in bounded domains and introduces techniques (domain extensions and weighted spaces) that may extend to more general geometries.

Abstract

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in [Coron J.M., Math. Control Signals Systems, 5(3) (1992) 295--312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.

Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

TL;DR

This work studies global exact boundary controllability for ideal incompressible magnetohydrodynamic flows in a planar rectangle, with boundary controls active on the two vertical walls. By reformulating the MHD system in Elsässer variables and applying Coron's return method, the authors prove global exact controllability for the Elsässer system and then transfer the result to the original MHD system through a pressure-like corrector, addressing the challenging nonlinear coupling. A fixed-point scheme in a weighted, extended-domain framework, together with carefully constructed initial-data extensions, establishes a local null controllability result that is then scaled and glued to obtain the global, small-time controllability. The approach provides a first rigorous step toward boundary controllability of ideal MHD in bounded domains and introduces techniques (domain extensions and weighted spaces) that may extend to more general geometries.

Abstract

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in [Coron J.M., Math. Control Signals Systems, 5(3) (1992) 295--312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.

Paper Structure

This paper contains 10 sections, 10 theorems, 92 equations, 1 figure.

Table of Contents

  1. Main results and outline of the work
  2. Related literature and outline of the proofs
  3. Preparations and notations
  4. Proofs of the main results
  5. Proof of the local null controllability result
  6. Definition of a fixed point space
  7. Construction of a fixed point map
  8. Existence of a fixed point
  9. Proof of Lemma \ref{['lemma:stepsForContraction']} (1): $F$ is a self map of $X_{\nu,k}$
  10. Proof of Lemma \ref{['lemma:stepsForContraction']} (2): $F$ is contractive with respect to $\|\cdot\|_X$

Key Result

Theorem 1.1

Let the integer $\tilde{m} \geq 3$ and the control time $T > 0$ be fixed. Then, for all initial- and final data $(\mathbf{z}^+_0, \mathbf{z}^-_0, \mathbf{z}^+_T, \mathbf{z}^-_T) \in {\rm C}^{\tilde{m},\alpha}_{\sigma,\Gamma_0}(\overline{\Omega}; \mathbb{R}^2)^4$, there exists a solution $(\mathbf{z}

Figures (1)

  • Figure 1: A sketch of the rectangular domain $\Omega$ and its extensions, while the arrows indicate $\overline{\mathbf{y}}(\mathbf{x},t)$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 10 more