Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. Application to the Navier-Stokes system - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2009

Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. Application to the Navier-Stokes system

Résumé

Let $A : \mathcal{D}(A)\to \mathcal{X}$ be the generator of an analytic semigroup and $B : \mathcal{U} \to [{\cal D}(A^*)]'$ a quasi-bounded operator. In this paper, we consider the stabilization of the system $y'=Ay+Bu$ where $u$ is the linear combination of a family $(v_1,\ldots,v_K)$. Our main result shows that if $(A^*,B^*)$ satisfies a unique continuation property and if $K$ is greater or equal to the maximum of the geometric multiplicities of the the unstable modes of $A$, then the system is generically stabilizable with respect to the family $(v_1,\ldots,v_K)$. With the same functional framework, we also prove the stabilizability of a class of nonlinear system when using feedback or dynamical controllers. We apply these results to stabilize the Navier--Stokes equations in 2D and in 3D by using boundary control.
Fichier principal
Vignette du fichier
StabNSDimFinie038.pdf (351.57 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00431041 , version 1 (10-11-2009)
hal-00431041 , version 2 (08-11-2010)

Identifiants

  • HAL Id : hal-00431041 , version 1

Citer

Mehdi Badra, Takéo Takahashi. Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. Application to the Navier-Stokes system. 2009. ⟨hal-00431041v1⟩
272 Consultations
307 Téléchargements

Partager

Gmail Mastodon Facebook X LinkedIn More