Spectral analysis of discrete elliptic operators and applications in control theory

Damien Allonsius (1) , Franck Boyer (2, 3) , Morgan Morancey (1)

1 I2M - Institut de Mathématiques de Marseille
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 IUF - Institut universitaire de France

In this paper we propose an analysis of discrete spectral properties for a finite difference discretization of quite general 1D second-order self-adjoint elliptic operators. We particularly investigate some (uniform in the discretization parameter) qualitative behavior of eigenfunctions and eigenvalues. With those estimates we manage to obtain new results for the construction of bounded families of controls for semi-discrete parabolic PDEs, in particular for boundary controls of a coupled parabolic system with fewer controls than equations.

Mots clés: moments method; control theory; Discrete spectral analysis; systems of parabolic equations; Sturm-Liouville operator

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Damien Allonsius, Franck Boyer, Morgan Morancey. Spectral analysis of discrete elliptic operators and applications in control theory. Numerische Mathematik, 2018, 140 (4), pp.857--911. ⟨10.1007/s00211-018-0983-1⟩. ⟨hal-01422168v2⟩