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 and Continuous Dynamical Systems, 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. 

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[36]

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

[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. 

[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.

[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.

[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.

[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. 

[42]

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

[43]

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

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. 

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[36]

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

[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. 

[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.

[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.

[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.

[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. 

[42]

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

[43]

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

Figure 1.  The fluid-plate system
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