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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 |
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. |
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), 420-463.
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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