# American Institute of Mathematical Sciences

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 and 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), 420-463. doi: 10.1137/090778146. [2] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM COCV, 15 (2009), 934-968. doi: 10.1051/cocv:2008059. [3] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control and Optimization, 48 (2009), 1797-1830. doi: 10.1137/070682630. [4] V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optimization, 50 (2012), 2288-2307. doi: 10.1137/110837164. [5] V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering, Springer-Verlag, London, 2011. [6] V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1-20. doi: 10.1137/09077607X. [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, Internat. Ser. Numer. Math., 155, Birkhaüser, Basel, 2007, 13-46. doi: 10.1007/978-3-7643-7721-2_2. [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), 2704-2746. doi: 10.1016/j.na.2005.09.012. [9] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), 1-145. doi: 10.1090/memo/0852. [10] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445. [11] V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197-205. doi: 10.1051/cocv:2003009. [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, Springer, 2006. [13] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, The Univ. of Chicago Press, Chicago, IL, 1988. [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), 289-314. doi: 10.3934/dcds.2004.10.289. [15] A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control, J. of Math. Fluid Mechanics, 3 (2001), 259-301. doi: 10.1007/PL00000972. [16] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Transl. of Math. Mongraphs, 187, AMS, Providence, Rhode Island, 2000. [17] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Nonlinear steady problems, volume 39 of Springer Tracts in Natural Philosophy. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [18] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. 1, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-5364-8. [19] C. Grandmont, B. Maury and A. Soualah, Multiscale modelling of the respiratory tract: A theoretical framework, ESAIM: Proc., 23 (2008), 10-29. doi: 10.1051/proc:082302. [20] J. L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 2002. [21] S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback, Int. J. Numer. Anal. Model, 4 (2007), 608-624. [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), 1159-1187. doi: 10.3934/dcds.2010.27.1159. [23] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669. doi: 10.1016/j.matpur.2007.04.002. [24] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828. doi: 10.1137/050628726. [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), 1119-1122. [26] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Amer. Math. Soc., Providence, RI, 2001.

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##### 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), 420-463. doi: 10.1137/090778146. [2] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM COCV, 15 (2009), 934-968. doi: 10.1051/cocv:2008059. [3] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control and Optimization, 48 (2009), 1797-1830. doi: 10.1137/070682630. [4] V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optimization, 50 (2012), 2288-2307. doi: 10.1137/110837164. [5] V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering, Springer-Verlag, London, 2011. [6] V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1-20. doi: 10.1137/09077607X. [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, Internat. Ser. Numer. Math., 155, Birkhaüser, Basel, 2007, 13-46. doi: 10.1007/978-3-7643-7721-2_2. [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), 2704-2746. doi: 10.1016/j.na.2005.09.012. [9] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), 1-145. doi: 10.1090/memo/0852. [10] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445. [11] V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197-205. doi: 10.1051/cocv:2003009. [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, Springer, 2006. [13] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, The Univ. of Chicago Press, Chicago, IL, 1988. [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), 289-314. doi: 10.3934/dcds.2004.10.289. [15] A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of boundary feedback control, J. of Math. Fluid Mechanics, 3 (2001), 259-301. doi: 10.1007/PL00000972. [16] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Transl. of Math. Mongraphs, 187, AMS, Providence, Rhode Island, 2000. [17] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Nonlinear steady problems, volume 39 of Springer Tracts in Natural Philosophy. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [18] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. 1, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-5364-8. [19] C. Grandmont, B. Maury and A. Soualah, Multiscale modelling of the respiratory tract: A theoretical framework, ESAIM: Proc., 23 (2008), 10-29. doi: 10.1051/proc:082302. [20] J. L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 2002. [21] S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback, Int. J. Numer. Anal. Model, 4 (2007), 608-624. [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), 1159-1187. doi: 10.3934/dcds.2010.27.1159. [23] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669. doi: 10.1016/j.matpur.2007.04.002. [24] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828. doi: 10.1137/050628726. [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), 1119-1122. [26] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Amer. Math. Soc., Providence, RI, 2001.
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