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Approximate and null controllability of a parabolic system with coupling terms of order one

Amélie Dupouy

Abstract

We study two notions of controllability on a parabolic system with coupling terms of order one. Based on existing results on, on one side parabolic systems with coupling terms of order zero, and on the other one parabolic systems with coupling terms of order one where the control domain is an interval, we give here some controllability conditions in the case where the coupling term is of order one and the control domain is not necessarily an interval.

Approximate and null controllability of a parabolic system with coupling terms of order one

Abstract

We study two notions of controllability on a parabolic system with coupling terms of order one. Based on existing results on, on one side parabolic systems with coupling terms of order zero, and on the other one parabolic systems with coupling terms of order one where the control domain is an interval, we give here some controllability conditions in the case where the coupling term is of order one and the control domain is not necessarily an interval.

Paper Structure

This paper contains 14 sections, 11 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Presentation of the problem
  3. Two-component cascade case
  4. Spectral properties of the adjoint operator and main tools
  5. Spectrum, eigenfunctions and biorthogonal basis
  6. Useful tools
  7. Approximate controllability
  8. Already existing results with no order one coupling term
  9. Back to our system
  10. A necessary and sufficient condition
  11. A particular case
  12. Null controllability
  13. Known results without order one coupling term
  14. Moments method with coupling term of order one

Key Result

Theorem 1.1

Let us suppose that $p \in W^{1,\infty}(0,\pi) \: \cap \: W^{2,\infty}(\omega)$, $q \in L^{\infty}(0,\pi) \: \cap \: W^{1,\infty}(\omega)$ and Then the system is null controllable at any time T.

Figures (1)

  • Figure 1: Example of choices for $p_0$, $\theta_a$ and $\theta_b$

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 1
  • Proposition 2.1
  • Definition 2
  • proof
  • Theorem 2.1
  • Definition 3
  • Theorem 2.2
  • ...and 16 more