Eigenvalues Distributions and Control Theory

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.