Spectral conditions for admissibility and observability of Schrödinger systems: Applications to finite element discretizations

Article type: Research Article

Authors: Ervedoza, Sylvain

Affiliations: Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des États Unis, 78035 Versailles Cedex, France. Tel.: +33 1 39 25 36 16; Fax: +33 1 39 25 46 45; E-mail: [email protected]

Abstract: In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of iż=A0z, where A0 is an unbounded self-adjoint positive definite operator with compact resolvent. In order to address this problem, we present several spectral criteria for admissibility and observability of such systems, which will be used to derive several results for space semi-discretizations of iż=A0z. Our approach provides very general results, which stand in any dimension and for any regular mesh (in the sense of finite elements). We also present applications to admissibility and observability for fully discrete approximation schemes, and to controllability and stabilization issues.

Keywords: observability, admissibility, Schrödinger equation, finite element method, resolvent estimates, controllability, stabilization

DOI: 10.3233/ASY-2010-1028

Journal: Asymptotic Analysis, vol. 71, no. 1-2, pp. 1-32, 2011