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Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

Manuel González-Burgos, Lydia Ouaili

Abstract

The general goal of this work is to obtain upper and lower bounds for the $L^2$-norm of biorthogonal families to complex exponential functions associated to sequences $\{ Λ_k \}_{k \ge 1} \subset \mathbb C$ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $T > 0$: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $T$ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence.

Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

Abstract

The general goal of this work is to obtain upper and lower bounds for the -norm of biorthogonal families to complex exponential functions associated to sequences which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time : a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence.
Paper Structure (16 sections, 37 theorems, 410 equations)

This paper contains 16 sections, 37 theorems, 410 equations.

Table of Contents

  1. Introduction and main results
  2. Some general properties of sequences under the assumptions of Definition \ref{['d1']}. Some examples
  3. Proof of the first main result
  4. An infinite product
  5. Additional properties and proof of Theorem \ref{['lemme technique']}
  6. A lower bound for the norm of arbitrary biorthogonal families: Proof of Theorem \ref{['estimbas famillbio']}
  7. A lower bound for the norm of arbitrary biorthogonal families. First part
  8. A lower bound for the norm of arbitrary biorthogonal families. Second part
  9. Application to the boundary controllability problem for some parabolic systems
  10. A $2 \times 2$ linear coupled parabolic system
  11. The linear phase-field system
  12. Proof of Propositions \ref{['p6']}, \ref{['p3']}, \ref{['p4']} and \ref{['p5']}
  13. Proof of Proposition \ref{['p6']}
  14. Proof of Proposition \ref{['p3']}
  15. Proof of Proposition \ref{['p4']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Lambda = \left\{\Lambda_k\right\} _{k \ge 1} \subset \mathbb{C}\xspace$ be a sequence satisfying assumptions item1--item5, in Definition d1, the gap condition and ($\mathcal{N}$ is the counting function associated with the sequence $\Lambda$, defined in counting), for some parameters $\beta \in [0, \infty)$, $\rho , p, \alpha \in (0, \infty)$ and $q \in \mathbb{N}\xspace$. Then, there exi

Theorems & Definitions (81)

  • Definition 1.1
  • Theorem 1.1: BBGBO
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.2
  • Remark 1.6
  • Theorem 1.3
  • Remark 1.7
  • ...and 71 more