May  2017, 37(5): 2315-2373. doi: 10.3934/dcds.2017102

Feedback boundary stabilization of 2d fluid-structure interaction systems

1. 

CNRS / Univ Pau & Pays Adour, Laboratoire de Mathématiques et, de leurs Applications de Pau -Fédération IPRA, UMR5142, 64000, Pau, France

2. 

Institut Élie Cartan, UMR 7502, INRIA, Nancy-Université, CNRS, BP239, 54506 Vandœuvre-lès-Nancy Cedex, France, Team SPHINX. INRIA Nancy -Grand Est, 615, rue du Jardin Botanique, 54600 Villers-lès-Nancy, France

* Corresponding author: Mehdi Badra

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: The authors are partially supported by the project ANR IFSMACS (ANR-15-CE40-0010) financed by the French Agence Nationale de la Recherche

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the fluid.

Citation: Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102
References:
[1]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.   Google Scholar

[2]

M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.  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 Optim., 48 (2009), 1797-1830.  doi: 10.1137/070682630.  Google Scholar

[4]

M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst., 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.  Google Scholar

[5]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamic controllers. Application to Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463.  doi: 10.1137/090778146.  Google Scholar

[6]

M. Badra and T. Takahashi, Feedback stabilization of a fluid-rigid body interaction system, Adv. Differential Equations, 19 (2014), 1137-1184.   Google Scholar

[7]

M. Badra and T. Takahashi, Feedback stabilization of a simplified 1d fluid-particle system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 369-389.  doi: 10.1016/j.anihpc.2013.03.009.  Google Scholar

[8]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.  Google Scholar

[9]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations Mem. Amer. Math. Soc. , 181 (2006), x+128. doi: 10.1090/memo/0852.  Google Scholar

[10]

H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52.   Google Scholar

[11]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc. , Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[12]

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813.  doi: 10.1016/j.anihpc.2008.02.004.  Google Scholar

[13]

M. Boulakia and S. Guerrero, Regular solutions of a problem coupling a compressible fluid and an elastic structure, J. Math. Pures Appl.(9), 94 (2010), 341-365.  doi: 10.1016/j.matpur.2010.04.002.  Google Scholar

[14]

M. BoulakiaE. L. Schwindt and T. Takahashi, Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid, Interfaces Free Bound., 14 (2012), 273-306.  doi: 10.4171/IFB/282.  Google Scholar

[15]

M. Boulakia, Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech., 9 (2007), 262-294.  doi: 10.1007/s00021-005-0201-7.  Google Scholar

[16]

A. ChambolleB. DesjardinsM. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[17]

S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15.  Google Scholar

[18]

S. Court, Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part Ⅰ: The linearized system, Evol. Equ. Control Theory, 3 (2014), 59-82.  doi: 10.3934/eect.2014.3.59.  Google Scholar

[19]

S. Court, Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part Ⅱ: The nonlinear system, Evol. Equ. Control Theory, 3 (2014), 83-118.  doi: 10.3934/eect.2014.3.83.  Google Scholar

[20]

B. DesjardinsM. J. EstebanC. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538.  doi: 10.5209/rev_REMA.2001.v14.n2.17030.  Google Scholar

[21]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de {S}tokes, Comm. Partial Differential Equations, 21 (1996), 573-596.  doi: 10.1080/03605309608821198.  Google Scholar

[22]

H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad., 46 (1970), 1141-1143.  doi: 10.3792/pja/1195526510.  Google Scholar

[23]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259-301.  doi: 10.1007/PL00000972.  Google Scholar

[24]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.   Google Scholar

[25]

P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63.   Google Scholar

[26]

P. Grisvard, Elliptic Problems in Nonsmooth Domains vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar

[27]

P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures Appl.(9), 45 (1966), 207-290.   Google Scholar

[28]

J. Lequeurre, Null controllability of a fluid-structure system, SIAM J. Control Optim., 51 (2013), 1841-1872.  doi: 10.1137/110839163.  Google Scholar

[29]

J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., 43 (2011), 389-410.  doi: 10.1137/10078983X.  Google Scholar

[30]

J. Lequeurre, Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation, J. Math. Fluid Mech., 15 (2013), 249-271.  doi: 10.1007/s00021-012-0107-0.  Google Scholar

[31]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems vol. 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations vol. ~44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. Google Scholar

[33]

A. QuarteroniM. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods, Computing and Visualization in Science, 2 (2000), 163-197.  doi: 10.1007/s007910050039.  Google Scholar

[34]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl.(9), 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.  Google Scholar

[35]

J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.  doi: 10.1016/j.anihpc.2006.06.008.  Google Scholar

[36]

J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.  doi: 10.1137/080744761.  Google Scholar

[37]

J.-P. Raymond and T. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Issue in Discrete and Continuous Dynamical Systems A, 27 (2010), 1159-1187.   Google Scholar

[38]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic).  doi: 10.1137/050628726.  Google Scholar

[39]

J.-P. Raymond, Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1537-1564.  doi: 10.3934/dcdsb.2010.14.1537.  Google Scholar

[40]

J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling the Navier-Stokes equations and the Lamé system, J. Math. Pures Appl.(9), 102 (2014), 546-596.  doi: 10.1016/j.matpur.2013.12.004.  Google Scholar

[41]

T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532.   Google Scholar

[42]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis North-Holland Publishing Co. , Amsterdam, 1977, Studies in Mathematics and its Applications, Vol. 2. Google Scholar

[43]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators vol. 18 of North-Holland Mathematical Library, North-Holland Publishing Co. , Amsterdam-New York, 1978. Google Scholar

show all references

References:
[1]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.   Google Scholar

[2]

M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.  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 Optim., 48 (2009), 1797-1830.  doi: 10.1137/070682630.  Google Scholar

[4]

M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst., 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.  Google Scholar

[5]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamic controllers. Application to Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463.  doi: 10.1137/090778146.  Google Scholar

[6]

M. Badra and T. Takahashi, Feedback stabilization of a fluid-rigid body interaction system, Adv. Differential Equations, 19 (2014), 1137-1184.   Google Scholar

[7]

M. Badra and T. Takahashi, Feedback stabilization of a simplified 1d fluid-particle system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 369-389.  doi: 10.1016/j.anihpc.2013.03.009.  Google Scholar

[8]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.  Google Scholar

[9]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations Mem. Amer. Math. Soc. , 181 (2006), x+128. doi: 10.1090/memo/0852.  Google Scholar

[10]

H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52.   Google Scholar

[11]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc. , Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[12]

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813.  doi: 10.1016/j.anihpc.2008.02.004.  Google Scholar

[13]

M. Boulakia and S. Guerrero, Regular solutions of a problem coupling a compressible fluid and an elastic structure, J. Math. Pures Appl.(9), 94 (2010), 341-365.  doi: 10.1016/j.matpur.2010.04.002.  Google Scholar

[14]

M. BoulakiaE. L. Schwindt and T. Takahashi, Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid, Interfaces Free Bound., 14 (2012), 273-306.  doi: 10.4171/IFB/282.  Google Scholar

[15]

M. Boulakia, Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech., 9 (2007), 262-294.  doi: 10.1007/s00021-005-0201-7.  Google Scholar

[16]

A. ChambolleB. DesjardinsM. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[17]

S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.  doi: 10.2140/pjm.1989.136.15.  Google Scholar

[18]

S. Court, Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part Ⅰ: The linearized system, Evol. Equ. Control Theory, 3 (2014), 59-82.  doi: 10.3934/eect.2014.3.59.  Google Scholar

[19]

S. Court, Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part Ⅱ: The nonlinear system, Evol. Equ. Control Theory, 3 (2014), 83-118.  doi: 10.3934/eect.2014.3.83.  Google Scholar

[20]

B. DesjardinsM. J. EstebanC. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538.  doi: 10.5209/rev_REMA.2001.v14.n2.17030.  Google Scholar

[21]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de {S}tokes, Comm. Partial Differential Equations, 21 (1996), 573-596.  doi: 10.1080/03605309608821198.  Google Scholar

[22]

H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad., 46 (1970), 1141-1143.  doi: 10.3792/pja/1195526510.  Google Scholar

[23]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259-301.  doi: 10.1007/PL00000972.  Google Scholar

[24]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.   Google Scholar

[25]

P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63.   Google Scholar

[26]

P. Grisvard, Elliptic Problems in Nonsmooth Domains vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar

[27]

P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures Appl.(9), 45 (1966), 207-290.   Google Scholar

[28]

J. Lequeurre, Null controllability of a fluid-structure system, SIAM J. Control Optim., 51 (2013), 1841-1872.  doi: 10.1137/110839163.  Google Scholar

[29]

J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., 43 (2011), 389-410.  doi: 10.1137/10078983X.  Google Scholar

[30]

J. Lequeurre, Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation, J. Math. Fluid Mech., 15 (2013), 249-271.  doi: 10.1007/s00021-012-0107-0.  Google Scholar

[31]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems vol. 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations vol. ~44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. Google Scholar

[33]

A. QuarteroniM. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods, Computing and Visualization in Science, 2 (2000), 163-197.  doi: 10.1007/s007910050039.  Google Scholar

[34]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl.(9), 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.  Google Scholar

[35]

J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.  doi: 10.1016/j.anihpc.2006.06.008.  Google Scholar

[36]

J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.  doi: 10.1137/080744761.  Google Scholar

[37]

J.-P. Raymond and T. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Issue in Discrete and Continuous Dynamical Systems A, 27 (2010), 1159-1187.   Google Scholar

[38]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic).  doi: 10.1137/050628726.  Google Scholar

[39]

J.-P. Raymond, Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1537-1564.  doi: 10.3934/dcdsb.2010.14.1537.  Google Scholar

[40]

J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling the Navier-Stokes equations and the Lamé system, J. Math. Pures Appl.(9), 102 (2014), 546-596.  doi: 10.1016/j.matpur.2013.12.004.  Google Scholar

[41]

T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8 (2003), 1499-1532.   Google Scholar

[42]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis North-Holland Publishing Co. , Amsterdam, 1977, Studies in Mathematics and its Applications, Vol. 2. Google Scholar

[43]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators vol. 18 of North-Holland Mathematical Library, North-Holland Publishing Co. , Amsterdam-New York, 1978. Google Scholar

Figure 1.  The fluid-plate system
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