Spectral characterization of weak observability for infinite-dimensional evolution systems

Keywords :

• Équations d'évolution
• Théorie spectrale
• Inégalité de résolvante
• Problèmes inverses
• Perturbation relativement compacte.

This thesis studies exact and weak observability for infinite-dimensional evolutionary systems generated by anti-adjoint operators, with a particular focus on relatively compact perturbations. First, we establish sufficient conditions guaranteeing the preservation of the exact observability of a system under such perturbations. This analysis relies on a detailed spectral study of the perturbed generator and on the generalized Rouché theorem, which ensures the stability of the eigenvalues ​​and their multiplicities. We then develop a spectral characterization of weak observability by introducing the notion of spectral coercivity of the observation operator, equivalent to a resolvent inequality. This framework allows us to obtain Hautus-type observability estimates relating the properties of the resolvent of an elliptic operator to the observability of the system. Finally, we show that weak observability is equivalent to a spectral condition guaranteeing that the energy of the observed eigenmodes does not decay too rapidly with respect to their frequencies. These results are applied to the wave equation, particularly on the rectangle and the disk, illustrating the scope of this approach.

Directors:

• Faouzi Triki (Université Grenoble Alpes )

Reporters:

• Ali Wehbe (Université Libanaise )
• Pierre Maréchal (Université de Toulouse )

Examinators:

• Anna Doubova (Université de Séville )
• Eric Bonnetier (Université Grenoble Alpes )