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May  2018, 23(3): 1107-1132. doi: 10.3934/dcdsb.2018144

## Feedback stabilization of a linear hydro-elastic system

 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27607, USA 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

* Corresponding author: lvbociu@ncsu.edu

Received  March 2017 Revised  August 2017 Published  February 2018

Fund Project: The first author is partially supported by NSF Grant DMS-1312801. The third author is partially supported by NSF Grant DMS-1616425.

It is known that the linear Stokes-Lamé system can be stabilized by a boundary feedback in the form of a dissipative velocity matching on the common interface [5]. Here we consider feedback stabilization for a generalized linear fluid-elasticity interaction, where the matching conditions on the interface incorporate the curvature of the common boundary and thus take into account the geometry of the problem. Such a coupled system is semigroup well-posed on the natural finite energy space [13], however, the system is not dissipative to begin with, which represents a key departure from the feedback control analysis in [5]. We prove that a damped version of the general linear hydro-elasticity model is exponentially stable. First, such a result is given for boundary dissipation of the form used in [5]. This proof resolves a more complex version, compared to the classical case, of the weighted energy methods, and addresses the lack of over-determination in the associated unique continuation result. The second theorem demonstrates how assumptions can be relaxed if a viscous damping is added in the interior of the solid.

Citation: Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144
##### References:
 [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electron. J. Differential Equations, 2000 (2000), 15 pp. (electronic). Google Scholar [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, in Fluids and Waves, vol. 440 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 15-54. Google Scholar [3] G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method, Appl. Math. (Warsaw), 35 (2008), 259-280. doi: 10.4064/am35-3-2. Google Scholar [4] G. Avalos and D. Toundykov, On stability and trace regularity of solutions to ReissnerMindlin-Timoshenko equations, in Modern Aspects of the Theory of Partial Differential Equations (eds. M. Ruzhansky and J. Wirth), vol. 216 of Operator Theory: Advances and Applications, Birkhäuser, 2011, 79-91. Google Scholar [5] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Equ., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9. Google Scholar [6] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. Google Scholar [7] G. Avalos and R. Triggiani, Backwards uniqueness of the C0-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system, Trans. Amer. Math. Soc., 362 (2010), 3535-3561. doi: 10.1090/S0002-9947-10-04851-8. Google Scholar [8] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and Waves, vol. 440 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 55-82. Google Scholar [9] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284. Google Scholar [10] V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of {N}avier-{S}tokes equations by high-and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746. doi: 10.1016/j.na.2005.09.012. Google Scholar [11] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055. Google Scholar [12] H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, Journal of Mathematical Fluid Mechanics, 6 (2004), 21-52. doi: 10.1007/s00021-003-0082-5. Google Scholar [13] L. Bociu, D. Toundykov and J.-P. Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction, SIAM J. Math. Anal., 47 (2015), 1958-2000. doi: 10.1137/140970689. Google Scholar [14] L. Bociu and J. -P. Zol´esio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, in Modern Aspects of the Theory of Partial Differential Equations, vol. 216 of Oper. Theory Adv. Appl., Birkh¨auser/Springer Basel AG, Basel, 2011, 93-120. 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] F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Contin. Dyn. Syst., 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557. Google Scholar [17] J. Chazarain and A. Piriou, Caractérisation des problémes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237. doi: 10.5802/aif.438. Google Scholar [18] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113. doi: 10.1137/0319008. Google Scholar [19] I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509. Google Scholar [20] I. Chueshov, I. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314. doi: 10.1007/s10884-009-9132-y. Google Scholar [21] P. G. Ciarlet, Mathematical Elasticity. Vol. I, vol. 20 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988, Three-dimensional elasticity. Google Scholar [22] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. Google Scholar [23] H. Cohen and S. I. Rubinow, Some mathematical topics in biology, in Proc. Symp. on System Theory, Polytechnic Press, New York, 1965,321-337.Google Scholar [24] C. Conca, J. Planchard, B. Thomas and R. Dautray, Problèmes Mathématiques en Couplage Fluide-Structure: Applications Aux Faisceaux Tubulaires, Editions Eyrolles, Paris, 1994.Google Scholar [25] C. Conca, J. San Martín H. and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. Google Scholar [26] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2. Google Scholar [27] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3 (1972), 428-445. doi: 10.1137/0503042. Google Scholar [28] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136. Google Scholar [29] B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538. Google Scholar [30] J. Donea, S. Giuliani and J. Halleux, An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering, 33 (1982), 689-723. doi: 10.1016/0045-7825(82)90128-1. Google Scholar [31] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ. 3 (2003), 419-441, Dedicated to Philippe Bénilan, doi: 10.1007/s00028-003-0110-1. Google Scholar [32] M. A. Fernández and P. Le Tallec, Linear stability analysis in fluid-structure interaction with transpiration. Ⅰ. Formulation and mathematical analysis, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4805-4835. doi: 10.1016/j.cma.2003.07.001. Google Scholar [33] M. A. Fernández and P. Le Tallec, Linear stability analysis in fluid-structure interaction with transpiration. Ⅱ. Numerical analysis and applications, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4837-4873. doi: 10.1016/j.cma.2003.08.001. Google Scholar [34] M. A. Fernández and M. Moubachir, An exact block-Newton algorithm for solving fluid-structure interaction problems, C. R. Math. Acad. Sci. Paris, 336 (2003), 681-686. doi: 10.1016/S1631-073X(03)00151-1. Google Scholar [35] A. R. Galper and T. Miloh, Motion stability of a deformable body in an ideal fluid with applications to the N spheres problem, Phys. Fluids, 10 (1998), 119-130. doi: 10.1063/1.869570. Google Scholar [36] C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem, J. Math. Fluid Mech., 4 (2002), 76-94. doi: 10.1007/s00021-002-8536-9. Google Scholar [37] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159. Google Scholar [38] C. Grandmont and Y. Maday, Fluid-structure interaction: A theoretical point of view, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 1-22. Google Scholar [39] M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954. Google Scholar [40] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648. Google Scholar [41] M. A. Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl., 223 (1998), 126-150. doi: 10.1006/jmaa.1998.5963. Google Scholar [42] M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11 pp. Google Scholar [43] T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9. Google Scholar [44] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467. Google Scholar [45] M. Ikawa, A mixed problem for hyperbolic equations of second order with non-homogeneous Neumann type boundary condition, Osaka J. Math., 6 (1969), 339-374. Google Scholar [46] V. Komornik, Exact Controllability and Stabilization, Masson, Paris, 1994. Google Scholar [47] J. E. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. Google Scholar [48] J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10, Society for Industrial and Applied Mathematics, SIAM; Philadelphia, PA, 1989. Google Scholar [49] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201. Google Scholar [50] I. Lasiecka, Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions, Appl. Anal., 47 (1992), 191-212. doi: 10.1080/00036819208840140. Google Scholar [51] I. Lasiecka, Mathematical Control Theory of Coupled PDEs, vol. 75, Society for Industrial and Applied Mathematics, SIAM; Philadelphia, PA, 2002. Google Scholar [52] I. Lasiecka and Y. Lu, Interface feedback control stabilization of a nonlinear fluid-structure interaction, Nonlinear Anal., 75 (2012), 1449-1460. doi: 10.1016/j.na.2011.04.018. Google Scholar [53] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. Google Scholar [54] I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations, Partial Differential Equation Methods in Control and Shape Analysis (Pisa); Lecture Notes in Pure and Appl. Math. , Dekker, 188 (1997), 215-243. Google Scholar [55] J. -L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués. Tome 1, vol. 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988, Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. Google Scholar [56] W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations, in Control Problems for Systems Described by Partial Differential Equations and Applications (Gainesville, Fla., 1986), vol. 97 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, 1987,307-312. Google Scholar [57] Y. Lu, Uniform decay rates for the energy in nonlinear fluid structure interaction with monotone viscous damping, Palest. J. Math., 2 (2013), 215-232. Google Scholar [58] Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model, Nonlinear Anal. Real World Appl., 25 (2015), 51-63. doi: 10.1016/j.nonrwa.2015.02.006. Google Scholar [59] C. S. Morawetz, J. V. Ralston and W. A. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math., 30 (1977), 447-508. doi: 10.1002/cpa.3160300405. Google Scholar [60] B. Palmerio, A two-dimensional FEM adaptive moving-node method for steady Euler flow simulations, Computer Methods in Applied Mechanics and Engineering, 71 (1988), 315-340. doi: 10.1016/0045-7825(88)90038-2. Google Scholar [61] S. Piperno and C. Farhat, Design of efficient partitioned procedures for the transient solution of aeroelastic problems, in Fluid-Structure Interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 23-49. Google Scholar [62] J. P. Quinn and D. L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 97-127. doi: 10.1017/S0308210500018072. Google Scholar [63] R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge, 1982, Translated from the Japanese by Katsumi Miyahara. Google Scholar [64] J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. Google Scholar [65] J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1959,125-263. Google Scholar [66] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77. doi: 10.1007/s00021-003-0083-4. Google Scholar [67] N. Takashi and T. J. Hughes, An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body, Computer Methods in Applied Mechanics and Engineering, 95 (1992), 115-138. doi: 10.1016/0045-7825(92)90085-X. Google Scholar [68] P. L. Tallec and J. Mouro, Fluid structure interaction with large structural displacements, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 3039-3067, Advances in Computational Methods for Fluid-Structure Interaction. doi: 10.1016/S0045-7825(00)00381-9. Google Scholar [69] J. Tambača, M. Kosor, S. Čanić and D. Paniagua, Mathematical modeling of vascular stents, SIAM J. Appl. Math., 70 (2010), 1922-1952. doi: 10.1137/080722618. Google Scholar [70] D. Tataru, A priori estimates of carleman's type in domains with boundary, J. Math. Pures Appl. (9), 73 (1994), 355-387. Google Scholar [71] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl., 75 (1996), 367-408. Google Scholar [72] R. Triggiani and P.-F. Yao, Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim., 46 (2002), 331-375, Special issue dedicated to the memory of Jacques-Louis Lions. doi: 10.1007/s00245-002-0751-5. Google Scholar

show all references

##### References:
 [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electron. J. Differential Equations, 2000 (2000), 15 pp. (electronic). Google Scholar [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, in Fluids and Waves, vol. 440 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 15-54. Google Scholar [3] G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method, Appl. Math. (Warsaw), 35 (2008), 259-280. doi: 10.4064/am35-3-2. Google Scholar [4] G. Avalos and D. Toundykov, On stability and trace regularity of solutions to ReissnerMindlin-Timoshenko equations, in Modern Aspects of the Theory of Partial Differential Equations (eds. M. Ruzhansky and J. Wirth), vol. 216 of Operator Theory: Advances and Applications, Birkhäuser, 2011, 79-91. Google Scholar [5] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Equ., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9. Google Scholar [6] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. Google Scholar [7] G. Avalos and R. Triggiani, Backwards uniqueness of the C0-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system, Trans. Amer. Math. Soc., 362 (2010), 3535-3561. doi: 10.1090/S0002-9947-10-04851-8. Google Scholar [8] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and Waves, vol. 440 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 55-82. Google Scholar [9] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284. Google Scholar [10] V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of {N}avier-{S}tokes equations by high-and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746. doi: 10.1016/j.na.2005.09.012. Google Scholar [11] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055. Google Scholar [12] H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, Journal of Mathematical Fluid Mechanics, 6 (2004), 21-52. doi: 10.1007/s00021-003-0082-5. Google Scholar [13] L. Bociu, D. Toundykov and J.-P. Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction, SIAM J. Math. Anal., 47 (2015), 1958-2000. doi: 10.1137/140970689. Google Scholar [14] L. Bociu and J. -P. Zol´esio, Linearization of a coupled system of nonlinear elasticity and viscous fluid, in Modern Aspects of the Theory of Partial Differential Equations, vol. 216 of Oper. Theory Adv. Appl., Birkh¨auser/Springer Basel AG, Basel, 2011, 93-120. 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] F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Contin. Dyn. Syst., 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557. Google Scholar [17] J. Chazarain and A. Piriou, Caractérisation des problémes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237. doi: 10.5802/aif.438. Google Scholar [18] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113. doi: 10.1137/0319008. Google Scholar [19] I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent, Discrete Contin. Dyn. Syst., 20 (2008), 459-509. Google Scholar [20] I. Chueshov, I. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314. doi: 10.1007/s10884-009-9132-y. Google Scholar [21] P. G. Ciarlet, Mathematical Elasticity. Vol. I, vol. 20 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988, Three-dimensional elasticity. Google Scholar [22] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. Google Scholar [23] H. Cohen and S. I. Rubinow, Some mathematical topics in biology, in Proc. Symp. on System Theory, Polytechnic Press, New York, 1965,321-337.Google Scholar [24] C. Conca, J. Planchard, B. Thomas and R. Dautray, Problèmes Mathématiques en Couplage Fluide-Structure: Applications Aux Faisceaux Tubulaires, Editions Eyrolles, Paris, 1994.Google Scholar [25] C. Conca, J. San Martín H. and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25 (2000), 1019-1042. Google Scholar [26] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352. doi: 10.1007/s00205-005-0385-2. Google Scholar [27] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3 (1972), 428-445. doi: 10.1137/0503042. Google Scholar [28] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71. doi: 10.1007/s002050050136. Google Scholar [29] B. Desjardins, M. J. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model, Rev. Mat. Complut., 14 (2001), 523-538. Google Scholar [30] J. Donea, S. Giuliani and J. Halleux, An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering, 33 (1982), 689-723. doi: 10.1016/0045-7825(82)90128-1. Google Scholar [31] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ. 3 (2003), 419-441, Dedicated to Philippe Bénilan, doi: 10.1007/s00028-003-0110-1. Google Scholar [32] M. A. Fernández and P. Le Tallec, Linear stability analysis in fluid-structure interaction with transpiration. Ⅰ. Formulation and mathematical analysis, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4805-4835. doi: 10.1016/j.cma.2003.07.001. Google Scholar [33] M. A. Fernández and P. Le Tallec, Linear stability analysis in fluid-structure interaction with transpiration. Ⅱ. Numerical analysis and applications, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4837-4873. doi: 10.1016/j.cma.2003.08.001. Google Scholar [34] M. A. Fernández and M. Moubachir, An exact block-Newton algorithm for solving fluid-structure interaction problems, C. R. Math. Acad. Sci. Paris, 336 (2003), 681-686. doi: 10.1016/S1631-073X(03)00151-1. Google Scholar [35] A. R. Galper and T. Miloh, Motion stability of a deformable body in an ideal fluid with applications to the N spheres problem, Phys. Fluids, 10 (1998), 119-130. doi: 10.1063/1.869570. Google Scholar [36] C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem, J. Math. Fluid Mech., 4 (2002), 76-94. doi: 10.1007/s00021-002-8536-9. Google Scholar [37] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159. Google Scholar [38] C. Grandmont and Y. Maday, Fluid-structure interaction: A theoretical point of view, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 1-22. Google Scholar [39] M. D. Gunzburger, H.-C. Lee and G. A. Seregin, Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2 (2000), 219-266. doi: 10.1007/PL00000954. Google Scholar [40] K.-H. Hoffmann and V. N. Starovoitov, On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., 9 (1999), 633-648. Google Scholar [41] M. A. Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl., 223 (1998), 126-150. doi: 10.1006/jmaa.1998.5963. Google Scholar [42] M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control, 8 (1998), 11 pp. Google Scholar [43] T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9. Google Scholar [44] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467. Google Scholar [45] M. Ikawa, A mixed problem for hyperbolic equations of second order with non-homogeneous Neumann type boundary condition, Osaka J. Math., 6 (1969), 339-374. Google Scholar [46] V. Komornik, Exact Controllability and Stabilization, Masson, Paris, 1994. Google Scholar [47] J. E. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. Google Scholar [48] J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10, Society for Industrial and Applied Mathematics, SIAM; Philadelphia, PA, 1989. Google Scholar [49] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201. Google Scholar [50] I. Lasiecka, Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions, Appl. Anal., 47 (1992), 191-212. doi: 10.1080/00036819208840140. Google Scholar [51] I. Lasiecka, Mathematical Control Theory of Coupled PDEs, vol. 75, Society for Industrial and Applied Mathematics, SIAM; Philadelphia, PA, 2002. Google Scholar [52] I. Lasiecka and Y. Lu, Interface feedback control stabilization of a nonlinear fluid-structure interaction, Nonlinear Anal., 75 (2012), 1449-1460. doi: 10.1016/j.na.2011.04.018. Google Scholar [53] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. Google Scholar [54] I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations, Partial Differential Equation Methods in Control and Shape Analysis (Pisa); Lecture Notes in Pure and Appl. Math. , Dekker, 188 (1997), 215-243. Google Scholar [55] J. -L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués. Tome 1, vol. 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988, Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. Google Scholar [56] W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations, in Control Problems for Systems Described by Partial Differential Equations and Applications (Gainesville, Fla., 1986), vol. 97 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, 1987,307-312. Google Scholar [57] Y. Lu, Uniform decay rates for the energy in nonlinear fluid structure interaction with monotone viscous damping, Palest. J. Math., 2 (2013), 215-232. Google Scholar [58] Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model, Nonlinear Anal. Real World Appl., 25 (2015), 51-63. doi: 10.1016/j.nonrwa.2015.02.006. Google Scholar [59] C. S. Morawetz, J. V. Ralston and W. A. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math., 30 (1977), 447-508. doi: 10.1002/cpa.3160300405. Google Scholar [60] B. Palmerio, A two-dimensional FEM adaptive moving-node method for steady Euler flow simulations, Computer Methods in Applied Mechanics and Engineering, 71 (1988), 315-340. doi: 10.1016/0045-7825(88)90038-2. Google Scholar [61] S. Piperno and C. Farhat, Design of efficient partitioned procedures for the transient solution of aeroelastic problems, in Fluid-Structure Interaction, Innov. Tech. Ser., Kogan Page Sci., London, 2003, 23-49. Google Scholar [62] J. P. Quinn and D. L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 97-127. doi: 10.1017/S0308210500018072. Google Scholar [63] R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, Cambridge, 1982, Translated from the Japanese by Katsumi Miyahara. Google Scholar [64] J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161 (2002), 113-147. doi: 10.1007/s002050100172. Google Scholar [65] J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1959,125-263. Google Scholar [66] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech., 6 (2004), 53-77. doi: 10.1007/s00021-003-0083-4. Google Scholar [67] N. Takashi and T. J. Hughes, An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body, Computer Methods in Applied Mechanics and Engineering, 95 (1992), 115-138. doi: 10.1016/0045-7825(92)90085-X. Google Scholar [68] P. L. Tallec and J. Mouro, Fluid structure interaction with large structural displacements, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 3039-3067, Advances in Computational Methods for Fluid-Structure Interaction. doi: 10.1016/S0045-7825(00)00381-9. Google Scholar [69] J. Tambača, M. Kosor, S. Čanić and D. Paniagua, Mathematical modeling of vascular stents, SIAM J. Appl. Math., 70 (2010), 1922-1952. doi: 10.1137/080722618. Google Scholar [70] D. Tataru, A priori estimates of carleman's type in domains with boundary, J. Math. Pures Appl. (9), 73 (1994), 355-387. Google Scholar [71] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl., 75 (1996), 367-408. Google Scholar [72] R. Triggiani and P.-F. Yao, Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim., 46 (2002), 331-375, Special issue dedicated to the memory of Jacques-Louis Lions. doi: 10.1007/s00245-002-0751-5. Google Scholar
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