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Tracking controllability on moving targets for parabolic equations

Jone Apraiz, Jon Asier Bárcena-Petisco, Judit Muñoz-Matute

Abstract

In this paper, we study the tracking controllability of a 1D parabolic type equation. Notably, with controls acting on the boundary, we seek to approximately control the solution of the equation on specific points of the domain. We prove that acting on one boundary point, we control the solution on one target point, whereas acting on two boundary points, we can control the solution on up to two target points. In order to do so, when the target is fixed, we study the controllability by minimizing the corresponding problem with duality results. Afterwards, we study the controllability on moving points by applying a transformation that takes the problem to a fixed target. Lastly, we also solve some of these control problems numerically and compute approximations of the solutions and the desired targets, which validates our theoretical methodology.

Tracking controllability on moving targets for parabolic equations

Abstract

In this paper, we study the tracking controllability of a 1D parabolic type equation. Notably, with controls acting on the boundary, we seek to approximately control the solution of the equation on specific points of the domain. We prove that acting on one boundary point, we control the solution on one target point, whereas acting on two boundary points, we can control the solution on up to two target points. In order to do so, when the target is fixed, we study the controllability by minimizing the corresponding problem with duality results. Afterwards, we study the controllability on moving points by applying a transformation that takes the problem to a fixed target. Lastly, we also solve some of these control problems numerically and compute approximations of the solutions and the desired targets, which validates our theoretical methodology.

Paper Structure

This paper contains 13 sections, 4 theorems, 87 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Tracking controllability on fixed target points
  4. Controllability of trajectories on moving target points
  5. Controlling one target point
  6. Controlling two target points
  7. Numerical results
  8. Discretization and optimization strategy
  9. Example 1: One boundary control and one tracked point for the heat equation
  10. Example 2: Two boundary controls and two tracked points for the heat equation
  11. Example 3: One boundary control and one tracked point for variable coefficients
  12. Example 4: One boundary control and one moving tracked point
  13. Open problems and possible extensions

Key Result

Lemma 3.2

Let $x_1,\ldots,x_N\in (0,L)$ satisfying $x_1<\cdots<x_N$. Then, con:heat0gen is approximately controllable on $(0,T)\times\{x_1,\ldots,x_N\}$ if and only if Moreover, if con:heat0gen is approximately controllable on $(0,T)\times\{x_1,\ldots,x_N\}$, given $\varepsilon>0$ and $(w_1,\ldots,w_N)\in (L^2(0,T))^N$, we can define the control that allows to obtain for $(f_1,\ldots,f_N)$ the minimizer

Figures (7)

  • Figure 1: Target $w(t)$ and achieved trace $y(t,x_1)$ at $x_1=0.5$ for $\varepsilon=10^{-1}$ (left) and $\varepsilon=10^{-2}$ (right).
  • Figure 2: Boundary control $v(t)=\partial_x p(t,1)$ for $\varepsilon=10^{-1}$ (left) and $\varepsilon=10^{-2}$ (right).
  • Figure 3: Target $w_1(t)$ and achieved trace $y(t,x_1)$ at $x_1=0.25$ (left), and target $w_2(t)$ and achieved trace $y(t,x_2)$ at $x_2=0.5$ (right), for $\varepsilon=10^{-3}$.
  • Figure 4: Boundary control $v_0(t)=\partial_x p(t,0)$ (left) and boundary control $v_1(t)=\partial_x p(t,1)$ (right), for $\varepsilon=10^{-3}$.
  • Figure 5: Target $w(t)$ and achieved trace $y(t,x_1)$ at $x_1=0.75$ (left) and boundary control $v(t)=\partial_x p(t,L)$ (right) for $\varepsilon=10^{-3}$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.2: Duality for interior pointwise controllability
  • proof : Proof of Lemma \ref{['lm:interiordualgen']}
  • Theorem 3.3: Simultaneous interior pointwise control
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 1 more