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On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations

Enrique Fernández-Cara, Arnaud Münch, Diego A. Souza

TL;DR

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier–Stokes equations with Dirichlet boundary conditions with mixed finite element Lagrangian approximations that are relatively easy to handle.

Abstract

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation, see~[A.V. Fursikov, O.Yu. Imanuvilov: {\it Controllability of Evolutions Equations,} Lectures Notes Series, Vol.~34, Seoul National University, 1996]; this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables $x_1$, $x_2$ and $t$ that is second--order in time and fourth--order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.

On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations

TL;DR

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier–Stokes equations with Dirichlet boundary conditions with mixed finite element Lagrangian approximations that are relatively easy to handle.

Abstract

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation, see~[A.V. Fursikov, O.Yu. Imanuvilov: {\it Controllability of Evolutions Equations,} Lectures Notes Series, Vol.~34, Seoul National University, 1996]; this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables , and that is second--order in time and fourth--order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.
Paper Structure (20 sections, 15 theorems, 152 equations, 18 figures, 3 tables)

This paper contains 20 sections, 15 theorems, 152 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction. The controllability problems
  2. A strategy for the computation of null controls for the heat equation
  3. First mixed formulation with modified variables
  4. Second mixed formulation with modified variables
  5. A reformulation of \ref{['m_hh']}
  6. A numerical approximation based on Lagrangian finite elements
  7. The Arrow-Hurwicz algorithm
  8. A numerical experiment
  9. A strategy for the computation of null controls for the Stokes equations
  10. A first mixed formulation of \ref{['pb-psigma-s']}
  11. A second mixed formulation of \ref{['pb-psigma-s']}
  12. A third mixed reformulation of \ref{['pb-psigma-mm']} with an additional multiplier
  13. Another formulation related to \ref{['reform_pb-psigma-s']}
  14. A fifth (and final) mixed formulation
  15. A numerical approximation of \ref{['pb-psigma-mmm']} (without justification)
  16. ...and 5 more sections

Key Result

Theorem 1

The heat equation heat is null-controllable at any time $T>0$.

Figures (18)

  • Figure 1: The domain and the mesh. Number of vertices: 2 800. Number of elements (tetrahedra): 14 094. Total number of variables: 20 539.
  • Figure 2: $\omega=(0.2,0.6)$; $y_0(\mathbf{x})=1000$. Cuts at $x_1=0.28$ and $x_1=0.52$ of the control $v_h$ ( Left) and the state ( Right).
  • Figure 3: Evolution of the state: $t=0.2$ and $t=0.6$ ( Left), $t=0.4$ and $t=0.8$ ( Right).
  • Figure 4: Evolution of the $L^2$ norms of the control and the state.
  • Figure 5: $\omega=(0.2,0.6)$; $\mathbf{y}_0(\mathbf{x})=(1000,0)$. Cuts of $v_{1,h}$ and $10^{10} v_{2,h}$ at $x_1=0.28$ ( Left) and $x_1=0.52$ ( Right).
  • ...and 13 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Proposition 1
  • Theorem 5
  • Proposition 2
  • proof
  • Remark 2
  • ...and 13 more