March  2014, 3(1): 147-166. doi: 10.3934/eect.2014.3.147

Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method

1. 

UFR de Sciences Appliquées et Technologie, Université Gaston Berger, B.P. 234 Saint-Louis, Senegal, Senegal

2. 

Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, blvd du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Received  June 2013 Revised  November 2013 Published  February 2014

In this work we study the exponential stabilization of the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$, around a given steady-state flow, by means of a boundary control. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions. The resulting feedback control is proven to be globally exponentially stabilizing the steady states of the two and three-dimensional Navier-Stokes equations.
Citation: Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
References:
[1]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite-dimensional feedback or dynamical controllers: Application to the Navier-Stokes system,, SIAM J. Control and Optimization, 49 (2011), 420. doi: 10.1137/090778146. Google Scholar

[2]

M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system,, ESAIM COCV, 15 (2009), 934. doi: 10.1051/cocv:2008059. Google Scholar

[3]

M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control and Optimization, 48 (2009), 1797. doi: 10.1137/070682630. Google Scholar

[4]

V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers,, SIAM J. Control Optimization, 50 (2012), 2288. doi: 10.1137/110837164. Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering,, Springer-Verlag, (2011). Google Scholar

[6]

V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier-Stokes equations,, SIAM J. Control Optim., 49 (2011), 1. doi: 10.1137/09077607X. Google Scholar

[7]

V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, d = 2, 3, via feedback stabilization of its linearization,, in Control of Coupled Partial Differential Equations, (2007), 13. doi: 10.1007/978-3-7643-7721-2_2. Google Scholar

[8]

V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers,, Nonlinear Anal, 64 (2006), 2704. doi: 10.1016/j.na.2005.09.012. Google Scholar

[9]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006), 1. doi: 10.1090/memo/0852. Google Scholar

[10]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers,, Indiana Univ. Math. J., 53 (2004), 1443. doi: 10.1512/iumj.2004.53.2445. Google Scholar

[11]

V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM: Control, 9 (2003), 197. doi: 10.1051/cocv:2003009. Google Scholar

[12]

F. Boyer and P. Fabrie, Éléments D'analyse Pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles, Mathématiques et Applications,, vol. 52, (2006). Google Scholar

[13]

P. Constantin and C. Foias, Navier-Stokes Equations,, Chicago Lectures in Mathematics, (1988). Google Scholar

[14]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications,, Discrete and Cont. Dyn. Syst., 10 (2004), 289. doi: 10.3934/dcds.2004.10.289. Google Scholar

[15]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control,, J. of Math. Fluid Mechanics, 3 (2001), 259. doi: 10.1007/PL00000972. Google Scholar

[16]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications,, Transl. of Math. Mongraphs, (2000). Google Scholar

[17]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, Vol. II, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar

[18]

G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, vol. 1, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar

[19]

C. Grandmont, B. Maury and A. Soualah, Multiscale modelling of the respiratory tract: A theoretical framework,, ESAIM: Proc., 23 (2008), 10. doi: 10.1051/proc:082302. Google Scholar

[20]

J. L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires,, Dunod, (2002). Google Scholar

[21]

S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback,, Int. J. Numer. Anal. Model, 4 (2007), 608. Google Scholar

[22]

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two-dimensional Navier-Stokes equations with finite-dimensional controllers,, Discrete Contin. Dynam. Systems, 27 (2010), 1159. doi: 10.3934/dcds.2010.27.1159. Google Scholar

[23]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures Appl., 87 (2007), 627. doi: 10.1016/j.matpur.2007.04.002. Google Scholar

[24]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726. Google Scholar

[25]

A. Sene, B. A. Wane and D. Y. Le Roux, Control of irrigation channels with variable bathymetry and time dependent stabilization rate,, C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1119. Google Scholar

[26]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, Amer. Math. Soc., (2001). Google Scholar

show all references

References:
[1]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite-dimensional feedback or dynamical controllers: Application to the Navier-Stokes system,, SIAM J. Control and Optimization, 49 (2011), 420. doi: 10.1137/090778146. Google Scholar

[2]

M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system,, ESAIM COCV, 15 (2009), 934. doi: 10.1051/cocv:2008059. Google Scholar

[3]

M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control and Optimization, 48 (2009), 1797. doi: 10.1137/070682630. Google Scholar

[4]

V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers,, SIAM J. Control Optimization, 50 (2012), 2288. doi: 10.1137/110837164. Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering,, Springer-Verlag, (2011). Google Scholar

[6]

V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier-Stokes equations,, SIAM J. Control Optim., 49 (2011), 1. doi: 10.1137/09077607X. Google Scholar

[7]

V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, d = 2, 3, via feedback stabilization of its linearization,, in Control of Coupled Partial Differential Equations, (2007), 13. doi: 10.1007/978-3-7643-7721-2_2. Google Scholar

[8]

V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers,, Nonlinear Anal, 64 (2006), 2704. doi: 10.1016/j.na.2005.09.012. Google Scholar

[9]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006), 1. doi: 10.1090/memo/0852. Google Scholar

[10]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers,, Indiana Univ. Math. J., 53 (2004), 1443. doi: 10.1512/iumj.2004.53.2445. Google Scholar

[11]

V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM: Control, 9 (2003), 197. doi: 10.1051/cocv:2003009. Google Scholar

[12]

F. Boyer and P. Fabrie, Éléments D'analyse Pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles, Mathématiques et Applications,, vol. 52, (2006). Google Scholar

[13]

P. Constantin and C. Foias, Navier-Stokes Equations,, Chicago Lectures in Mathematics, (1988). Google Scholar

[14]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications,, Discrete and Cont. Dyn. Syst., 10 (2004), 289. doi: 10.3934/dcds.2004.10.289. Google Scholar

[15]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control,, J. of Math. Fluid Mechanics, 3 (2001), 259. doi: 10.1007/PL00000972. Google Scholar

[16]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications,, Transl. of Math. Mongraphs, (2000). Google Scholar

[17]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, Vol. II, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar

[18]

G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, vol. 1, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar

[19]

C. Grandmont, B. Maury and A. Soualah, Multiscale modelling of the respiratory tract: A theoretical framework,, ESAIM: Proc., 23 (2008), 10. doi: 10.1051/proc:082302. Google Scholar

[20]

J. L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires,, Dunod, (2002). Google Scholar

[21]

S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback,, Int. J. Numer. Anal. Model, 4 (2007), 608. Google Scholar

[22]

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two-dimensional Navier-Stokes equations with finite-dimensional controllers,, Discrete Contin. Dynam. Systems, 27 (2010), 1159. doi: 10.3934/dcds.2010.27.1159. Google Scholar

[23]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures Appl., 87 (2007), 627. doi: 10.1016/j.matpur.2007.04.002. Google Scholar

[24]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726. Google Scholar

[25]

A. Sene, B. A. Wane and D. Y. Le Roux, Control of irrigation channels with variable bathymetry and time dependent stabilization rate,, C. R. Acad. Sci. Paris Ser. I, 346 (2008), 1119. Google Scholar

[26]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, Amer. Math. Soc., (2001). Google Scholar

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