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Large-time optimal observation domain for linear parabolic systems

Idriss Mazari-Fouquer, Yannick Privat, Emmanuel Trélat

Abstract

Given a well-posed linear evolution system settled on a domain $Ω$ of $\mathbb{R}^d$, an observation subset $ω\subsetΩ$ and a time horizon $T$, the observability constant is defined as the largest possible nonnegative constant such that the observability inequality holds for the pair $(ω,T)$. In this article we investigate the large-time behavior of the observation domain that maximizes the observability constant over all possible measurable subsets of a given Lebesgue measure. We prove that it converges exponentially, as the time horizon goes to infinity, to a limit set that we characterize. The mathematical technique is new and relies on a quantitative version of the bathtub principle.

Large-time optimal observation domain for linear parabolic systems

Abstract

Given a well-posed linear evolution system settled on a domain of , an observation subset and a time horizon , the observability constant is defined as the largest possible nonnegative constant such that the observability inequality holds for the pair . In this article we investigate the large-time behavior of the observation domain that maximizes the observability constant over all possible measurable subsets of a given Lebesgue measure. We prove that it converges exponentially, as the time horizon goes to infinity, to a limit set that we characterize. The mathematical technique is new and relies on a quantitative version of the bathtub principle.
Paper Structure (24 sections, 7 theorems, 107 equations)

This paper contains 24 sections, 7 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Large-time behavior of maximizers
  3. Main results
  4. Examples and comments on the assumptions
  5. The Dirichlet-Laplacian operator.
  6. The Robin-Laplacian operator.
  7. The Dirichlet Stokes operator in the 2D unit disk.
  8. System of coupled heat equations in the cube.
  9. On the sharpness of $\mathbf{(H_{\boldsymbol{L}})}$: periodic Laplacian in the 1D torus.
  10. Comments on the strategy of proof
  11. Proofs
  12. Proof of Theorem \ref{['theo:sigma1']}
  13. Bang-bang property.
  14. Characterization of $\omega^*$.
  15. Uniqueness and regularity of $\omega^*$.
  16. ...and 9 more sections

Key Result

Theorem 1

Under $\mathbf{(H_1)}$, we have Under $\mathbf{(H_1)}$, $\mathbf{(H_2)}$ and $\mathbf{(H_{\boldsymbol{L}})}$, any solution $a^\star_T\in\overline{\mathcal{U}}_L$ of Problem maxCTa satisfies

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:m1202']}.
  • ...and 6 more