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Feedback stabilization of a simplified 1d fluid-particle system. (English) Zbl 1302.74057

Summary: 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.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35Q35 PDEs in connection with fluid mechanics
76D55 Flow control and optimization for incompressible viscous fluids
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
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[1] Badra, M.; Takahashi, T., Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamic controllers. Application to Navier-Stokes system, SIAM J. Control Optim., 49, 2, 420-463 (2011) · Zbl 1217.93137
[2] M. Badra, T. Takahashi, On Fattorini criterion for approximate controllability and stabilizability of parabolic equations, preprint. · Zbl 1292.93022
[3] Barbu, V.; Rodrigues, S. S.; Shirikyan, A., Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49, 4, 1454-1478 (2011) · Zbl 1231.35141
[4] Bensoussan, A.; Da Prato, G.; Delfour, M. C.; Mitter, S. K., Representation and control of infinite dimensional systems, (Systems & Control: Foundations & Applications (2007), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA) · Zbl 1117.93002
[5] Boulakia, M.; Guerrero, S., Local null controllability of a fluid-solid interaction problem in dimension 3, J. Eur. Math. Soc., 15, 3, 825-856 (2013) · Zbl 1264.35163
[6] Boulakia, M.; Osses, A., Local null controllability of a two-dimensional fluid-structure interaction problem, ESAIM Control Optim. Calc. Var., 14, 1, 1-42 (2008) · Zbl 1149.35068
[7] Chapouly, M., Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim., 48, 3, 1567-1599 (2009) · Zbl 1282.93050
[8] Chowdhury, S.; Ramaswamy, M. M.; Raymond, J.-P., Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension, SIAM J. Control Optim., 50, 5, 2959-2987 (2012) · Zbl 1257.93016
[9] Conca, C.; San Martín, H. J.; Tucsnak, M., Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25, 5-6, 1019-1042 (2000) · Zbl 0954.35135
[10] Desjardins, B.; Esteban, M. J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146, 1, 59-71 (1999) · Zbl 0943.35063
[11] Desjardins, B.; Esteban, M. J., On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations, 25, 7-8, 1399-1413 (2000) · Zbl 0953.35118
[12] Doubova, A.; Fernández-Cara, E., Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci., 15, 5, 783-824 (2005) · Zbl 1122.93008
[13] Ervedoza, S.; Glass, O.; Guerrero, S.; Puel, J.-P., Local exact controllability for the one-dimensional compressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 206, 1, 189-238 (2012) · Zbl 1387.93039
[14] Fernández-Cara, E.; Guerrero, S., Null controllability of the Burgers system with distributed controls, Systems Control Lett., 56, 5, 366-372 (2007) · Zbl 1130.93015
[15] Galdi, G. P., On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, (Handbook of Mathematical Fluid Dynamics, vol. I (2002), North-Holland: North-Holland Amsterdam), 653-791 · Zbl 1230.76016
[16] Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5 (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0585.65077
[17] Grandmont, C.; Maday, Y., Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34, 3, 609-636 (2000) · Zbl 0969.76017
[18] Guerrero, S.; Imanuvilov, O. Yu., Remarks on global controllability for the Burgers equation with two control forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24, 6, 897-906 (2007) · Zbl 1248.93024
[19] Gunzburger, M. D.; Lee, H.-C.; Seregin, G. A., Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2, 3, 219-266 (2000) · Zbl 0970.35096
[20] Imanuvilov, O.; Takahashi, T., Exact controllability of a fluid-rigid body system, J. Math. Pures Appl. (9), 87, 4, 408-437 (2007) · Zbl 1124.35056
[21] Kato, T., Perturbation Theory for Linear Operators, Classics in Mathematics (1995), Springer-Verlag: Springer-Verlag Berlin, Reprint of the 1980 edition · Zbl 0836.47009
[22] Liu, Y.; Takahashi, T.; Tucsnak, M., Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19, 1, 20-42 (2013) · Zbl 1270.35259
[23] Raymond, J.-P.; Thevenet, T., Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete Contin. Dyn. Syst. Ser. A, 27, 3, 1159-1187 (2010) · Zbl 1211.93103
[24] San Martín, J. A.; Starovoitov, V.; Tucsnak, M., Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., 161, 2, 113-147 (2002) · Zbl 1018.76012
[25] Silvestre, A. L., On the slow motion of a self-propelled rigid body in a viscous incompressible fluid, J. Math. Anal. Appl., 274, 1, 203-227 (2002) · Zbl 1121.76321
[26] Vázquez, J. L.; Zuazua, E., Large time behavior for a simplified 1D model of fluid-solid interaction, Comm. Partial Differential Equations, 28, 9-10, 1705-1738 (2003) · Zbl 1071.74017
[27] Vázquez, J. L.; Zuazua, E., Lack of collision in a simplified 1D model for fluid-solid interaction, Math. Models Methods Appl. Sci., 16, 5, 637-678 (2006) · Zbl 1387.35634
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