Noureddine Lamsahel – Mathematics and Decision

$\begin{quote} \begin{center} \textbf{Eigenvalues distribution and control theory} \end{center} \medskip This work deals with the isogeometric Galerkin discretization of the eigenvalues problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. The main objective is the use of GLT theory to study the behavior of the gap of discrete spectra toward the uniform gap condition needed for the uniform boundary observability/controllability problems. The analysis refers to a regular B-spline basis and concave or convex reparametrizations. Under suitable assumptions on the reparametrization transformation, we prove that structure emerges within the distribution of the eigenvalues once we reframe the problem into GLT-symbol analysis. We also demonstrate numerically, that the necessary average gap condition proposed in [1] is not equivalent to the uniform gap condition. However, by improving the result in [2] we construct sufficient criteria that guarantee the uniform gap property. \smallskip \begin{enumerate} \item[] [1] D. Bianchi and S. Serra-Capizzano. Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids. Calcolo, 55(4):1-28, 2018. \smallskip \item[] [2] D. Bianchi. Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators. Calcolo, 58(3):38, 2021. \end{enumerate} \end{quote}$