Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical null controllability of parabolic PDEs using Lagrangian methods

Enrique Fernandez-Cara, Roberto Morales, Diego A. Souza

Abstract

In this paper, we study several theoretical and numerical questions concerning the null controllability problems for linear parabolic equations and systems for several dimensions. The control is distributed and acts on a small subset of the domain. The main goal is to compute numerically a control that drives a numerical approximation of the state from prescribed initial data exactly to zero. We introduce a methodology for solving numerical controllability problems that is new in some sense. The main idea is to apply classical Lagrangian and Augmented Lagrangian techniques to suitable constrained extremal formulations that involve unbounded weights in time that make global Carleman inequalities possible. The theoretical results are validated by satisfactory numerical experiments for spatially 2D and 3D problems.

Numerical null controllability of parabolic PDEs using Lagrangian methods

Abstract

In this paper, we study several theoretical and numerical questions concerning the null controllability problems for linear parabolic equations and systems for several dimensions. The control is distributed and acts on a small subset of the domain. The main goal is to compute numerically a control that drives a numerical approximation of the state from prescribed initial data exactly to zero. We introduce a methodology for solving numerical controllability problems that is new in some sense. The main idea is to apply classical Lagrangian and Augmented Lagrangian techniques to suitable constrained extremal formulations that involve unbounded weights in time that make global Carleman inequalities possible. The theoretical results are validated by satisfactory numerical experiments for spatially 2D and 3D problems.

Paper Structure

This paper contains 20 sections, 5 theorems, 112 equations, 24 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature
  3. Plan of the paper
  4. Truncation of the extremal control problem
  5. A truncated problem
  6. Estimates of the convergence rate of the truncated solutions
  7. Solving the truncated extremal problem using Lagrange methods
  8. The Lagrangian
  9. The Augmented Lagrangian
  10. Some iterative algorithms
  11. Uzawa's algorithms
  12. Augmented Lagrangian methods
  13. Numerical null controllability of Stokes systems
  14. Numerical experiments
  15. Test #1: Controlling with $\omega$ small enough
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $\rho,\rho_0$ be the functions defined in Carleman:weights. Then Extremal:problem:01 has a unique minimizer $(y,v)\in \mathcal{Y}\times \mathcal{U}$. Moreover, we have the following characterization: where $p\in P$ is the unique solution to the variational (Lax-Milgram-like) problem

Figures (24)

  • Figure 1: Projected uncontrolled state at $x_1=0.35$ and $x_2=0.4$.
  • Figure 2: Evolution of the controlled state at $t=0,\, 0.0625,\, 0.0875$ and $0.15$ (From left to right).
  • Figure 3: Evolution of the controlled state at $t = 0.25,\, 0.375,\, 0.4375$ and $0.5$ (from left to right).
  • Figure 4: Projected uncontrolled state at $x_1=0.35$ (left) and $x_2=0.4$ (right).
  • Figure 5: The evolution of $\|\bar{y}(\cdot\,,t)\|_{L^2}$ and $\|y(\cdot\,,t)\|_{L^2}$ (left) and the evolution of $\| v(\cdot\,,t) \|_{L^2(\omega)}$ (right) as $t \in [0,T]$.
  • ...and 19 more figures

Theorems & Definitions (12)

  • Remark 1.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 3.1
  • ...and 2 more