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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
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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.

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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.

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V. BarbuI. 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.

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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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

V. BarbuZ. GrujićI. 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.

[10]

V. BarbuI. 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.

[11]

C. BardosG. 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.

[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.

[13]

L. BociuD. 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.

[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.

[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]

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.

[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.

[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.

[19]

I. ChueshovI. 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.

[20]

I. ChueshovI. 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.

[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.

[22]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.

[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.

[24]

C. Conca, J. Planchard, B. Thomas and R. Dautray, Problèmes Mathématiques en Couplage Fluide-Structure: Applications Aux Faisceaux Tubulaires, Editions Eyrolles, Paris, 1994.

[25]

C. ConcaJ. 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.

[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.

[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.

[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.

[29]

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.

[30]

J. DoneaS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

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Figure 1.  A 2D sample of an admissible control volume $\mathcal{D}$
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