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Feedback stabilization of a 3D fluid-structure model with a boundary control

Ndiaye, Moctar and Matignon, Denis and Raymond, Jean-Pierre Feedback stabilization of a 3D fluid-structure model with a boundary control. (2014) In: 21st International Symposium Mathematical Theory of Networks and Systems (MTNS 2014), 7 July 2014 - 11 July 2014 (Groningen, Netherlands).

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Abstract

We study a system coupling the incompressible Navier-Stokes equations in a 3D parallelepiped type domain with a damped plate equation. The plate is located in a part of the upper boundary of the fluid domain. The fluid domain depends on the deformation of the plate, and therefore it depends on time. We are interested in the stabilization, with a prescribed decay rate, of such a system in a neighborhood of a stationary solution, by a Dirichlet control acting at the boundary of the fluid domain. For that, we first study the stabilizability of the corresponding linearized system and we determine a finite-dimensional feedback control able to stabilize the linearized model. A crucial step in the analysis consists in showing that this linearized system can be rewritten thanks to an analytic semigroup, the infinitesimal generator of which has a compact resolvent. A fixed-point argument is used to prove the local stabilization of the original nonlinear system. The main difficulties come from the coupling between the fluid and plate equations, and the fact that the fluid domain varies with time, giving rise to geometric nonlinearities. The results of the paper may be adapted to other more complex geometrical configurations for the same type of system. Ongoing research concerns the numerics of the control problem.

Item Type:Invited Conference
Additional Information:Proceedings of 21st International Symposium Mathematical Theory of Networks and Systems (MTNS 2014). ISBN : 978-90-367-6321-9
Audience (conference):International conference proceedings
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National des Sciences Appliquées de Toulouse - INSA (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UPS (FRANCE)
Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)
Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE)
Laboratory name:
Funders:
ANR-project CISIFS ANR-09- BLAN-0213-03 - ANR-project HAMECMOPSYS ANR-11-BS03-0002
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Deposited By: Denis Matignon
Deposited On:29 Apr 2014 16:01

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