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Eigenvalues Distributions and Control Theory

N. Lamsahel, A. El Akri, A. Ratnani

TL;DR

This work analyzes the IgA discretization of the 1D Laplace eigenproblem under Dirichlet conditions through Generalized Locally Toeplitz (GLT) theory to understand eigenvalue distributions and gaps under reparametrizations. It establishes that eigenvalue ordering aligns with the GLT symbol, proves a uniform discrete Weyl law, and shows how convexity of the symbol and domain reparametrization affect spectral packing. A key outcome is a sufficient condition (bounded $m(n)$) guaranteeing a uniform gap, with linear $B$-splines ($p=1$) providing an explicit case and numerical results extending to higher degrees. Numerics reveal that the average spectral gap is not equivalent to the uniform gap, and propose reparametrizations, notably a modified $\Phi_p$, that yield the optimal uniform gap for a wide range of $p$. Overall, the paper provides a GLT-based framework to design discretizations that preserve observability in control problems and offers pathways to extend to higher dimensions and other discretization schemes.

Abstract

This work deals with the isogeometric Galerkin discretization of the eigenvalue problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory to study the behavior of the gap of discrete spectra toward the uniform gap condition needed for the uniform boundary observability/controllability problems. The analysis refers to a regular $B$-spline basis and concave or convex reparametrizations. Under suitable assumptions on the reparametrization transformation, we prove that structure emerges within the distribution of the eigenvalues once we reframe the problem into GLT-symbol analysis. We also demonstrate numerically, that the necessary average gap condition proposed in \cite{bianchi2018spectral} is not equivalent to the uniform gap condition. However, by improving the result in \cite{bianchi2021analysis} we construct sufficient criteria that guarantee the uniform gap property.

Eigenvalues Distributions and Control Theory

TL;DR

This work analyzes the IgA discretization of the 1D Laplace eigenproblem under Dirichlet conditions through Generalized Locally Toeplitz (GLT) theory to understand eigenvalue distributions and gaps under reparametrizations. It establishes that eigenvalue ordering aligns with the GLT symbol, proves a uniform discrete Weyl law, and shows how convexity of the symbol and domain reparametrization affect spectral packing. A key outcome is a sufficient condition (bounded ) guaranteeing a uniform gap, with linear -splines () providing an explicit case and numerical results extending to higher degrees. Numerics reveal that the average spectral gap is not equivalent to the uniform gap, and propose reparametrizations, notably a modified , that yield the optimal uniform gap for a wide range of . Overall, the paper provides a GLT-based framework to design discretizations that preserve observability in control problems and offers pathways to extend to higher dimensions and other discretization schemes.

Abstract

This work deals with the isogeometric Galerkin discretization of the eigenvalue problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory to study the behavior of the gap of discrete spectra toward the uniform gap condition needed for the uniform boundary observability/controllability problems. The analysis refers to a regular -spline basis and concave or convex reparametrizations. Under suitable assumptions on the reparametrization transformation, we prove that structure emerges within the distribution of the eigenvalues once we reframe the problem into GLT-symbol analysis. We also demonstrate numerically, that the necessary average gap condition proposed in \cite{bianchi2018spectral} is not equivalent to the uniform gap condition. However, by improving the result in \cite{bianchi2021analysis} we construct sufficient criteria that guarantee the uniform gap property.
Paper Structure (15 sections, 19 theorems, 159 equations, 14 figures, 3 tables)

This paper contains 15 sections, 19 theorems, 159 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Galerkin B-spline IgA Discretization
  4. Preliminaries on GLT Sequences
  5. The IgA GLT-Symbol
  6. IgA Eigenvalues Distribution Analysis
  7. Uniform Gap Condition
  8. The Case of Linear B-Splines
  9. Sufficient Condition
  10. Numerical tests
  11. The Case of Linear $B$-splines
  12. The Case of Quadratic $B$-splines
  13. The Case when $p \geq 3$
  14. Summary of Numerical Tests
  15. Conclusions and further comments

Key Result

Proposition 2.1

Let $(L_n)_{n\in\mathbb{N}^*} \sim_{\lambda} \omega$ with $\omega: [0,1] \times [0,\pi] \longrightarrow \mathbb{R}$ has a bounded essential range. Let $\xi$ be the monotone rearrangement of $\omega$. Then, we have

Figures (14)

  • Figure 1: The gap in function of $n$ ($p=1$).
  • Figure 2: The distribution of the distance between the square root eigenvalues for different values of $n$ and $\phi_1$.
  • Figure 3: The normalized eigenvalues and the symbol for different values of $n$ and for $\phi_1$.
  • Figure 4: The graph of $(m(n))_{n \geq 1}$ (left) and the gap in function of $n$ (right). Parameter values $p=2$, and $\phi=\phi_1$
  • Figure 5: The graph of $(m(n))_{n \geq 1}$ (left) and the gap in function of $n$ (right). Parameter values $p=2$, and $\phi=\phi_2$
  • ...and 9 more figures

Theorems & Definitions (36)

  • Definition 2.1: Spectral symbol
  • Definition 2.2: Outliers
  • Definition 2.3: Monotone rearrangement of the symbol
  • Proposition 2.1
  • Theorem 2.1: Discrete Weyl’s law , bianchi2021analysis
  • Corollary 2.1
  • Corollary 2.2
  • Remark 2.1
  • Theorem 2.2: garoni2017generalized, IgA GLT symbol
  • Corollary 2.3
  • ...and 26 more