Feedback stabilization of a simplified 1d fluid–particle system
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 369-389.

We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order $\sigma <{\sigma }_{0}$. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. 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 domains of the stationary state and of the stabilized solution are different.

DOI : https://doi.org/10.1016/j.anihpc.2013.03.009
Classification : 74F10,  35Q35,  76D55,  93C20,  93D15
Mots clés : Feedback stabilization, Fluid–structure interaction, Viscous Burgers equation
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author = {Badra, Mehdi and Takahashi, Tak\'eo},
title = {Feedback stabilization of a simplified 1d fluid{\textendash}particle system},
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Badra, Mehdi; Takahashi, Takéo. Feedback stabilization of a simplified 1d fluid–particle system. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 369-389. doi : 10.1016/j.anihpc.2013.03.009. http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.009/

[1] M. Badra, T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamic controllers. Application to Navier–Stokes system, SIAM J. Control Optim. 49 no. 2 (2011), 420-463 | MR 2784695 | Zbl 1217.93137

[2] M. Badra, T. Takahashi, On Fattorini criterion for approximate controllability and stabilizability of parabolic equations, preprint. | Numdam | MR 3264229

[3] V. Barbu, S.S. Rodrigues, A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier–Stokes equations, SIAM J. Control Optim. 49 no. 4 (2011), 1454-1478 | MR 2817486 | Zbl 1231.35141

[4] A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and control of infinite dimensional systems, Systems & Control: Foundations & Applications, second ed., Birkhäuser Boston Inc., Boston, MA (2007) | MR 2273323

[5] M. Boulakia, S. Guerrero, Local null controllability of a fluid–solid interaction problem in dimension 3, J. Eur. Math. Soc. 15 no. 3 (2013), 825-856, http://dx.doi.org/10.4171/JEMS/378 | EuDML 277776 | MR 3085093 | Zbl 1264.35163

[6] M. Boulakia, A. Osses, Local null controllability of a two-dimensional fluid–structure interaction problem, ESAIM Control Optim. Calc. Var. 14 no. 1 (2008), 1-42 | EuDML 245642 | Numdam | MR 2375750 | Zbl 1149.35068

[7] M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim. 48 no. 3 (2009), 1567-1599 | MR 2516179 | Zbl 1282.93050

[8] S. Chowdhury, M.M. Ramaswamy, J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier–Stokes system in one dimension, SIAM J. Control Optim. 50 no. 5 (2012), 2959-2987 | MR 3022094 | Zbl 1257.93016

[9] C. Conca, H.J. San Martín, M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25 no. 5–6 (2000), 1019-1042 | Zbl 0954.35135

[10] B. Desjardins, M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal. 146 no. 1 (1999), 59-71 | MR 1682663 | Zbl 0943.35063

[11] B. Desjardins, M.J. Esteban, On weak solutions for fluid–rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations 25 no. 7–8 (2000), 1399-1413 | MR 1765138 | Zbl 0953.35118

[12] A. Doubova, E. Fernández-Cara, Some control results for simplified one-dimensional models of fluid–solid interaction, Math. Models Methods Appl. Sci. 15 no. 5 (2005), 783-824 | MR 2139944 | Zbl 1122.93008

[13] S. Ervedoza, O. Glass, S. Guerrero, J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier–Stokes equation, Arch. Ration. Mech. Anal. 206 no. 1 (2012), 189-238 | MR 2968594 | Zbl 06102063

[14] E. Fernández-Cara, S. Guerrero, Null controllability of the Burgers system with distributed controls, Systems Control Lett. 56 no. 5 (2007), 366-372 | MR 2311198 | Zbl 1130.93015

[15] G.P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam (2002), 653-791 | MR 1942470 | Zbl 1230.76016

[16] V. Girault, P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics vol. 5, Springer-Verlag, Berlin (1986) | MR 851383 | Zbl 0413.65081

[17] C. Grandmont, Y. Maday, Existence for an unsteady fluid–structure interaction problem, M2AN Math. Model. Numer. Anal. 34 no. 3 (2000), 609-636 | EuDML 197391 | Numdam | MR 1763528 | Zbl 0969.76017

[18] S. Guerrero, O.Yu. Imanuvilov, Remarks on global controllability for the Burgers equation with two control forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 no. 6 (2007), 897-906 | EuDML 78768 | Numdam | MR 2371111 | Zbl 1248.93024

[19] M.D. Gunzburger, H.-C. Lee, 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 no. 3 (2000), 219-266 | Zbl 0970.35096

[20] O. Imanuvilov, T. Takahashi, Exact controllability of a fluid–rigid body system, J. Math. Pures Appl. (9) 87 no. 4 (2007), 408-437 | MR 2317341 | Zbl 1124.35056

[21] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin (1995) | MR 1335452 | Zbl 0836.47009

[22] Y. Liu, T. Takahashi, M. Tucsnak, Single input controllability of a simplified fluid–structure interaction model, ESAIM Control Optim. Calc. Var. 19 no. 1 (2013), 20-42 | EuDML 272791 | Numdam | MR 3023058 | Zbl 1270.35259

[23] J.-P. Raymond, T. Thevenet, Boundary feedback stabilization of the two dimensional Navier–Stokes equations with finite dimensional controllers, Discrete Contin. Dyn. Syst. Ser. A 27 no. 3 (2010), 1159-1187 | MR 2629581 | Zbl 1211.93103

[24] J.A. San Martín, V. Starovoitov, M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 no. 2 (2002), 113-147 | MR 1870954 | Zbl 1018.76012

[25] A.L. Silvestre, On the slow motion of a self-propelled rigid body in a viscous incompressible fluid, J. Math. Anal. Appl. 274 no. 1 (2002), 203-227 | MR 1936694 | Zbl 1121.76321

[26] J.L. Vázquez, E. Zuazua, Large time behavior for a simplified 1D model of fluid–solid interaction, Comm. Partial Differential Equations 28 no. 9–10 (2003), 1705-1738 | MR 2001181 | Zbl 1071.74017

[27] J.L. Vázquez, E. Zuazua, Lack of collision in a simplified 1D model for fluid–solid interaction, Math. Models Methods Appl. Sci. 16 no. 5 (2006), 637-678 | MR 2226121 | Zbl 05045353

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