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Stabilization and Best Actuator Location for the Navier--Stokes Equations

Airiau, Christophe and Buchot, Jean-Marie and Dubey, Ritesh Kumar and Fournié, Michel and Raymond, Jean-Pierre and Weller, Jessie Stabilization and Best Actuator Location for the Navier--Stokes Equations. (2017) SIAM Journal on Scientific Computing, 39 (5). B993-B1020. ISSN 1064-8275

(Document in English)

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Official URL: https://doi.org/10.1137/16M107503X


We study the numerical approximation of the boundary stabilization of the Navier--Stokes equations with mixed Dirichlet/Neumann boundary conditions, around an unstable stationary solution in a two dimensional domain. We first derive a semidiscrete controlled system, coming from a finite element approximation of the Navier--Stokes equations, which is new in the literature. We propose a new strategy for finding a boundary feedback control law able to stabilize the nonlinear semidiscrete controlled system in the presence of boundary disturbances. We determine the best control location. Next, we study the degree of stabilizability of the different real generalized eigenspaces of the controlled system. Based on that analysis, we determine an invariant subspace $\mathbb{Z}_u$ and the projection of the controlled system onto $\mathbb{Z}_u$. The projected system is used to determine feedback control laws. Our numerical results show that this control strategy is quite efficient when applied to the Navier--Stokes system for a Reynolds number $R_e=150$ with boundary perturbations.

Item Type:Article
Additional Information:Thanks to Society for Industrial and Applied Mathematics editor. The original PDF of the article can be found at : https://epubs.siam.org/doi/abs/10.1137/16M107503X
HAL Id:hal-01810784
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Other partners > SRM University (INDIA)
Laboratory name:
Agence Nationale de Recherche
Deposited On:08 Jun 2018 09:10

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