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Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems

F. Ammar Khodja, A. Benabdallah, M. González-Burgos, M. Morancey

Abstract

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time.

Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems

Abstract

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time.
Paper Structure (30 sections, 19 theorems, 201 equations)

This paper contains 30 sections, 19 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Presentation of the minimal null control time problem
  3. Main results
  4. A brief review of previous results
  5. Structure of the article
  6. Strategy of proof
  7. Localization of eigenfunctions
  8. Proof of Theorem \ref{['ThCondensationNulle']}
  9. Dealing with algebraically double eigenvalues
  10. Examples of application
  11. Pointwise controllability of a one dimensional heat equation
  12. A parabolic cascade system with a variable coupling coefficient: a particular case of internal control
  13. A parabolic cascade system with a variable coupling coefficient: boundary control
  14. Condensation of the spectrum
  15. Condensation of eigenvalues given by the Bohr index
  16. ...and 15 more sections

Key Result

Theorem 1.1

Assume that problem SystControlAbstrait is null controllable in time $T$. Then, there exists $C_T >0$ such that for any $y \in D(A^*)$, for any $\lambda \in \mathbb{C}$ with $\mathrm{Re}(\lambda) >0$, one has

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 50 more