Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system
ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 4, p. 934-968

We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.

DOI : https://doi.org/10.1051/cocv:2008059
Classification:  35Q30,  76D05,  76D07,  76D55,  93B52,  93C20,  93D15
Keywords: Navier-Stokes equation, feedback stabilization, Dirichlet control, Riccati equation
@article{COCV_2009__15_4_934_0,
     author = {Badra, Mehdi},
     title = {Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {4},
     year = {2009},
     pages = {934-968},
     doi = {10.1051/cocv:2008059},
     zbl = {pre05627163},
     mrnumber = {2567253},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2009__15_4_934_0}
}
Badra, Mehdi. Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 4, pp. 934-968. doi : 10.1051/cocv:2008059. http://www.numdam.org/item/COCV_2009__15_4_934_0/

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