We consider an electrically conducting 2-D channel fluid flow affected by a transverse magnetic field. The governing equations are the magnetohydrodynamics equations. We design an explicit finite-dimensional exponentially stabilizing feedback, given in a very simple form, easily manageable from the computational point of view, for the Hartmann−Poiseuille profile. Moreover, the stability is assured independently of the value of the magnetic Reynolds number. The control acts on the normal components of both velocity and magnetic field, on the upper wall only.

DOI: 10.1051/cocv/2016025

Keywords: Magnetohydrodynamics equations, Hartmann-Poiseuille profile, stabilization, feedback controller, eigenvalues

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@article{COCV_2017__23_4_1253_0, author = {Ionu\c{t} Munteanu}, title = {Boundary stabilization of a {2-D} periodic {MHD} channel flow, by proportional feedbacks}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1253--1266}, publisher = {EDP-Sciences}, volume = {23}, number = {4}, year = {2017}, doi = {10.1051/cocv/2016025}, zbl = {1375.76217}, mrnumber = {3716920}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016025/} }

TY - JOUR AU - Ionuţ Munteanu TI - Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1253 EP - 1266 VL - 23 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016025/ DO - 10.1051/cocv/2016025 LA - en ID - COCV_2017__23_4_1253_0 ER -

%0 Journal Article %A Ionuţ Munteanu %T Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1253-1266 %V 23 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016025/ %R 10.1051/cocv/2016025 %G en %F COCV_2017__23_4_1253_0

Ionuţ Munteanu. Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 4, pp. 1253-1266. doi : 10.1051/cocv/2016025. http://archive.numdam.org/articles/10.1051/cocv/2016025/

V. Barbu, Stabilization of Navier−Stokes flows. Springer, New York (2010). | MR | Zbl

Stabilization of a plane channel flow by noise wall normal controllers. Syst. Control Lett. 59 (2010) 608–614. | DOI | MR | Zbl

,Boundary Stabilization of Equilibrium Solutions to Parabolic Equations. IEEE Trans. Autom. Control 9 (2013) 2416–2420. | DOI | MR | Zbl

,D. Biskamp, Magnetohydrodynamical Turbulence. Cambridge University Press, Cambridge (2003). | Zbl

N. Bourbaki, Théories spectrales. Springer (2007).

P.A. Davidson, An Introduction to Magnetohydrodynamics. Cambridge University Press (2001). | MR | Zbl

Cross-helicity dynamo effect in magnetohydrodynamic turbulent channel flow. Phys. Plasmas 17 (2010) 012301. | DOI

and ,Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl. Danske Vidensk-abernes Selskab Mathematisk-fysiske Meddelelser XV (1937) 1–27.

,I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge Univ. Press, Cambridge, UK (2000). | MR | Zbl

Magnetohydrodynamic turbulent flow in a channel at low magnetic Reynolds number. J. Fluid Mech. 439 (2001) 367–394. | DOI | Zbl

and ,HB. Liu, P. Hu and I. Munteanu, Boundary feedback stabilization of Fisher’s equation. Preprint (2016). | arXiv | MR

L. Lixiang and E. Schuster, Heat Transfer Enhancement in 2D Magnetohydrodynamic Channel Flow by Boundary Feedback Control. 45th IEEE Conf. Decision and Control (2006) 5317–5322.

R. Moreau, Magnetohydrodynamics. Kluwer Academic Publishers (1990). | MR | Zbl

U. Muller and L. Buhler, Magneto fluid dynamics in Channels and Containers. Springer (2001).

Boundary feedback stabilization of periodic fluid flows in a magnetohydrodynamic channel. IEEE Tran. Autom. Control 58 (2013) 2119–2125. | DOI | MR | Zbl

,Normal feedback stabilization for linearized periodic MHD channel flow, at low magnetic Reynolds number. Syst. Control Lett. 62 (2013) 55–62. | DOI | MR | Zbl

,Stabilization of parabolic semilinear equations. Int. J. Control 90 (2017) 1063–1076. | DOI | MR | Zbl

,Boundary stabilization of the phase field system by finite-dimensional feedback controllers. J. Math. Anal. Appl. 412 (2014) 964–975. | DOI | MR | Zbl

,Boundary stabilization of the Navier−Stokes Equation with Fading Memory. Int. J. Control 88 (2015) 531–542. | DOI | MR | Zbl

,Stabilization of Semilinear Heat Equations, with Fading Memory, by boundary feedbacks. J. Differ. Equ. 259 (2015) 454-472. | DOI | MR | Zbl

,Stability of plane Hartmann flow subject to a transverse magnetic field. Phys. Fluids 16 (1973). | DOI | Zbl

and ,MHD channel flow control in 2D: Mixing enhancement by boundary feedback. Automatica 44 (2008) 2498–2507. | DOI | MR | Zbl

, and ,The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field. Fluid Dyn. Res. 17 (1996) 293–310. | DOI | MR | Zbl

,Magnetohydrodynamic state estimation with boundary sensors. Automatica 44 (2008) 2517–2527. | DOI | MR | Zbl

, and ,R. Vazquez, E. Schuster and M. Krstic, A closed-form full-state feedback controller for stabilization of 3D magnetohydrodynamic channel flow. ASME J. Dyn. Syst., Meas. Control 131 (2009).

Stabilization of linearized 2D magnetohydrodynamic channel flow by backstepping boundary control. Syst. Control Lett. 57 (2008) 805–812. | DOI | MR | Zbl

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