# American Institute of Mathematical Sciences

March  2015, 4(1): 89-106. doi: 10.3934/eect.2015.4.89

## Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller

 1 UFR de Sciences Appliquées et Technologie, Université Gaston Berger, B.P. 234 Saint-Louis 2 CEA-MITIC, Université Gaston Berger, B.P. 234 Saint-Louis, Senegal 3 Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, blvd du 11 novembre 1918, 69622 Villeurbanne Cedex

Received  August 2014 Revised  December 2014 Published  February 2015

This paper presents a global stabilization for the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$ around a given unstable equilibrium state, by means of a boundary normal feedback control. The control is expressed in terms of the velocity field by using a non-linear feedback law. In order to determine the feedback control law, we consider an extended system coupling the equations governing the perturbation with an equation satisfied by the control on the domain boundary. By using the Faedo-Galerkin method and a priori estimation techniques, a stabilizing boundary control is built. This control law ensures a decrease of the energy of the controlled discrete system. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions.
Citation: Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations and Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89
##### 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] V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optimization, 50 (2012), 2288-2307. doi: 10.1137/110837164. [4] V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series, Springer, London, 2011. doi: 10.1007/978-0-85729-043-4. [5] 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. [6] 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. [7] 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. [8] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128 pp. doi: 10.1090/memo/0852. [9] 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. [10] V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197-206. doi: 10.1051/cocv:2003009. [11] 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. doi: 10.1007/3-540-29819-3. [12] A. Diagne and A. Sene, Control of shallow water and sediment continuity coupled system, Math. Control Signals Syst., 25 (2013), 387-406. doi: 10.1007/s00498-012-0101-3. [13] 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. [14] 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. [15] A. V. Fursikov, M. Gunzburger, L. S. Hou and S. Manservisi, Optimal control for the Navier-Stokes equations, in Lectures on Applied Mathematics (eds. H.-J. Bungartz, R. H. W. Hoppe and C. Zenger), Springer, New York, 2000, 143-155. [16] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [17] M. S. Goudiaby, A. Sene and G. Kreiss, A delayed feedback control for network of open canals, Int. J. Dynam. Control, 1 (2013), 316-329. doi: 10.1007/s40435-013-0028-7. [18] M. S. Goudiaby, A. Sene and G. Kreiss, An algebraic approach for controlling cascade of reaches in irrigation canals, in Problems, Perspectives and Challenges of Agricultural Water Management (ed. M. Kumar), InTech, 2012, 369-390. [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.-W. He, R. Glowinski, R. Metcalfe, A. Nordlander and J. Periaux, Active control and drag optimization for flow past a circular cylinder, J. Comput. Phys., 163 (2000), 83-117. doi: 10.1006/jcph.2000.6556. [21] J. L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 2002. [22] E. M. D. Ngom, A. Sène and D. Y. Le Roux, Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method, Evolution Equations and Control Theory, 3 (2014), 147-166. doi: 10.3934/eect.2014.3.147. [23] D. S. Park, D. M. Ladd and E. W. Hendricks, Feedback control of von Karman vortex shedding behind a circular cylinder at low Reynolds numbers, Phys. Fluids, 6 (1994), 2390-2405. [24] S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback, Int. J. Numer. Anal. Model, 4 (2007), 608-624. [25] 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. [26] 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. [27] 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. [28] 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. doi: 10.1016/j.crma.2008.09.009.

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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] V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optimization, 50 (2012), 2288-2307. doi: 10.1137/110837164. [4] V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series, Springer, London, 2011. doi: 10.1007/978-0-85729-043-4. [5] 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. [6] 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. [7] 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. [8] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128 pp. doi: 10.1090/memo/0852. [9] 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. [10] V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197-206. doi: 10.1051/cocv:2003009. [11] 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. doi: 10.1007/3-540-29819-3. [12] A. Diagne and A. Sene, Control of shallow water and sediment continuity coupled system, Math. Control Signals Syst., 25 (2013), 387-406. doi: 10.1007/s00498-012-0101-3. [13] 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. [14] 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. [15] A. V. Fursikov, M. Gunzburger, L. S. Hou and S. Manservisi, Optimal control for the Navier-Stokes equations, in Lectures on Applied Mathematics (eds. H.-J. Bungartz, R. H. W. Hoppe and C. Zenger), Springer, New York, 2000, 143-155. [16] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [17] M. S. Goudiaby, A. Sene and G. Kreiss, A delayed feedback control for network of open canals, Int. J. Dynam. Control, 1 (2013), 316-329. doi: 10.1007/s40435-013-0028-7. [18] M. S. Goudiaby, A. Sene and G. Kreiss, An algebraic approach for controlling cascade of reaches in irrigation canals, in Problems, Perspectives and Challenges of Agricultural Water Management (ed. M. Kumar), InTech, 2012, 369-390. [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.-W. He, R. Glowinski, R. Metcalfe, A. Nordlander and J. Periaux, Active control and drag optimization for flow past a circular cylinder, J. Comput. Phys., 163 (2000), 83-117. doi: 10.1006/jcph.2000.6556. [21] J. L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 2002. [22] E. M. D. Ngom, A. Sène and D. Y. Le Roux, Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method, Evolution Equations and Control Theory, 3 (2014), 147-166. doi: 10.3934/eect.2014.3.147. [23] D. S. Park, D. M. Ladd and E. W. Hendricks, Feedback control of von Karman vortex shedding behind a circular cylinder at low Reynolds numbers, Phys. Fluids, 6 (1994), 2390-2405. [24] S. S. Ravindran, Stabilization of Navier-Stokes equations by boundary feedback, Int. J. Numer. Anal. Model, 4 (2007), 608-624. [25] 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. [26] 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. [27] 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. [28] 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. doi: 10.1016/j.crma.2008.09.009.
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