Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.

Previous Article
Optimal control for stochastic heat equation with memory
EECT Home
This Issue
Next Article
Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.

March 2014, 3(1): 59-82. doi: 10.3934/eect.2014.3.59

Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.

1.	Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France

Received March 2013 Revised November 2013 Published February 2014

Abstract
Related Papers

This paper is the first part of a work which consists in proving the stabilization to zero of a fluid-solid system, in dimension 2 and 3. The considered system couples a deformable solid and a viscous incompressible fluid which satisfies the incompressible Navier-Stokes equations. By deforming itself, the solid can interact with the environing fluid and then move itself. The control function represents nothing else than the deformation of the solid in its own frame of reference. We there prove that the velocities of the linearized system are stabilizable to zero with an arbitrary exponential decay rate, by a boundary deformation velocity which can be chosen in the form of a feedback operator. We then show that this boundary feedback operator can be obtained from an internal deformation of the solid which satisfies the linearized physical constraints that a self-propelled solid has to satisfy.

Keywords: boundary feedback stabilization., Fluid-structure interactions, Navier-Stokes equations, mechanics of deformable solids, Exponential stabilization.

Mathematics Subject Classification: Primary: 93C20, 35Q30, 76D05, 76D07, 74F10, 93C05, 93B52, 93D15; Secondary: 74A99, 35Q7.

Citation: Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

References:

[1]	G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Eq., 9 (2009), 341. doi: 10.1007/s00028-009-0015-9.
[2]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 2,, Birkhäuser, (1993).
[3]	T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid,, J. Nonlinear Sci., 21 (2011), 325. doi: 10.1007/s00332-010-9084-8.
[4]	T. Chambrion and A. Munnier, Generic controllability of 3d swimmers in a perfect fluid,, SIAM J. Control Optim., 50 (2012), 2814. doi: 10.1137/110828654.
[5]	S. Court, Existence of 3D strong solutions for a system modeling a deformable solid in a viscous incompressible fluid,, , ().
[6]	C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes,, (French) [Unique continuation of the solutions of the Stokes equation], 21 (1996), 573. doi: 10.1080/03605309608821198.
[7]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I,, Springer-Verlag, (1994).
[8]	O. Glass and L. Rosier, On the Control of the Motion of a Boat,, M3AS, (2011).
[9]	T. Kato, Perturbation Theory for Linear Operators,, Springer-Vermlag, (1995).
[10]	A. Y. Khapalov, Local controllability for a "swimming'' model,, SIAM J. Control Optim., 46 (2007), 655. doi: 10.1137/050638424.
[11]	A. Y. Khapalov, Geometric aspects of force controllability for a swimming model,, Appl. Math. Optim., 57 (2008), 98. doi: 10.1007/s00245-007-9013-x.
[12]	J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers,, Acta Appl. Math., 123* (2013), 175. doi: 10.1007/s10440-012-9760-9.*
[13]	A. Osses and J. P. Puel, Approximate controllability for a linear model of fluid structure interaction,, ESAIM Control Optim. Calc. Var., 4 (1999), 497. doi: 10.1051/cocv:1999119.
[14]	J. P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726.
[15]	J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921. doi: 10.1016/j.anihpc.2006.06.008.
[16]	J. P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398. doi: 10.1137/080744761.
[17]	J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms,, Quart. Appl. Math., 65 (2007), 405.
[18]	J. San Martín, J. F. Scheid, T. Takahashi and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming,, Arch. Ration. Mech. Anal., 188* (2008), 429. doi: 10.1007/s00205-007-0092-2.*
[19]	E. D. Sontag, Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition,, Springer-Verlag, (1998).
[20]	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.
[21]	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. doi: 10.1007/s00021-003-0083-4.
[22]	R. Temam, Problèmes Mathématiques en Plasticité,, Gauthier-Villars, (1983).
[23]	J. Zabczyk, Mathematical Control Theory: An Introduction,, Reprint of the 1995 edition. Modern Birkhäuser Classics. Birkhäuser Boston, (1995). doi: 10.1007/978-0-8176-4733-9.

show all references

References:

[1]	G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Eq., 9 (2009), 341. doi: 10.1007/s00028-009-0015-9.
[2]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 2,, Birkhäuser, (1993).
[3]	T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid,, J. Nonlinear Sci., 21 (2011), 325. doi: 10.1007/s00332-010-9084-8.
[4]	T. Chambrion and A. Munnier, Generic controllability of 3d swimmers in a perfect fluid,, SIAM J. Control Optim., 50 (2012), 2814. doi: 10.1137/110828654.
[5]	S. Court, Existence of 3D strong solutions for a system modeling a deformable solid in a viscous incompressible fluid,, , ().
[6]	C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes,, (French) [Unique continuation of the solutions of the Stokes equation], 21 (1996), 573. doi: 10.1080/03605309608821198.
[7]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I,, Springer-Verlag, (1994).
[8]	O. Glass and L. Rosier, On the Control of the Motion of a Boat,, M3AS, (2011).
[9]	T. Kato, Perturbation Theory for Linear Operators,, Springer-Vermlag, (1995).
[10]	A. Y. Khapalov, Local controllability for a "swimming'' model,, SIAM J. Control Optim., 46 (2007), 655. doi: 10.1137/050638424.
[11]	A. Y. Khapalov, Geometric aspects of force controllability for a swimming model,, Appl. Math. Optim., 57 (2008), 98. doi: 10.1007/s00245-007-9013-x.
[12]	J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers,, Acta Appl. Math., 123* (2013), 175. doi: 10.1007/s10440-012-9760-9.*
[13]	A. Osses and J. P. Puel, Approximate controllability for a linear model of fluid structure interaction,, ESAIM Control Optim. Calc. Var., 4 (1999), 497. doi: 10.1051/cocv:1999119.
[14]	J. P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726.
[15]	J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921. doi: 10.1016/j.anihpc.2006.06.008.
[16]	J. P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398. doi: 10.1137/080744761.
[17]	J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms,, Quart. Appl. Math., 65 (2007), 405.
[18]	J. San Martín, J. F. Scheid, T. Takahashi and M. Tucsnak, An initial and boundary value problem modeling of fish-like swimming,, Arch. Ration. Mech. Anal., 188* (2008), 429. doi: 10.1007/s00205-007-0092-2.*
[19]	E. D. Sontag, Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition,, Springer-Verlag, (1998).
[20]	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.
[21]	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. doi: 10.1007/s00021-003-0083-4.
[22]	R. Temam, Problèmes Mathématiques en Plasticité,, Gauthier-Villars, (1983).
[23]	J. Zabczyk, Mathematical Control Theory: An Introduction,, Reprint of the 1995 edition. Modern Birkhäuser Classics. Birkhäuser Boston, (1995). doi: 10.1007/978-0-8176-4733-9.

[1]	Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102
[2]	Andrei Fursikov, Alexey V. Gorshkov. Certain questions of feedback stabilization for Navier-Stokes equations. Evolution Equations & Control Theory, 2012, 1 (1) : 109-140. doi: 10.3934/eect.2012.1.109
[3]	Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89
[4]	Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
[5]	Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159
[6]	A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[7]	George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817
[8]	Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39
[9]	Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[10]	Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control & Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013
[11]	Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153
[12]	Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091
[13]	Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89
[14]	Assane Lo, Nasser-eddine Tatar. Exponential stabilization of a structure with interfacial slip. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6285-6306. doi: 10.3934/dcds.2016073
[15]	Mehdi 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 & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[16]	Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469
[17]	Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1
[18]	Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[19]	Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization. Networks & Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299
[20]	Kaïs Ammari, Denis Mercier. Boundary feedback stabilization of a chain of serially connected strings. Evolution Equations & Control Theory, 2015, 4 (1) : 1-19. doi: 10.3934/eect.2015.4.1

2017 Impact Factor: 1.049

American Institute of Mathematical Sciences