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

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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.

Sébastien Court ^1,

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
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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 and 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), 341-370. 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, Boston, Cambridge, MA, 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), 325-385. 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), 2814-2835. 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], Comm. Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.
[7]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Springer-Verlag, New York, 1994.
[8]	O. Glass and L. Rosier, On the Control of the Motion of a Boat, M3AS, 2011, to be published.
[9]	T. Kato, Perturbation Theory for Linear Operators, Springer-Vermlag, Berlin, 1995.
[10]	A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Control Optim., 46 (2007), 655-682. doi: 10.1137/050638424.
[11]	A. Y. Khapalov, Geometric aspects of force controllability for a swimming model, Appl. Math. Optim., 57 (2008), 98-124. 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), 175-200. 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), 497-513. 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), 790-828. 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), 921-951. 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), 5398-5443. 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), 405-424.
[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), 429-455. doi: 10.1007/s00205-007-0092-2.
[19]	E. D. Sontag, Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition, Springer-Verlag, New-York, 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), 1499-1532.
[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), 53-77. doi: 10.1007/s00021-003-0083-4.
[22]	R. Temam, Problèmes Mathématiques en Plasticité, Gauthier-Villars, Montrouge, 1983.
[23]	J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. 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), 341-370. 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, Boston, Cambridge, MA, 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), 325-385. 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), 2814-2835. 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], Comm. Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.
[7]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Springer-Verlag, New York, 1994.
[8]	O. Glass and L. Rosier, On the Control of the Motion of a Boat, M3AS, 2011, to be published.
[9]	T. Kato, Perturbation Theory for Linear Operators, Springer-Vermlag, Berlin, 1995.
[10]	A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Control Optim., 46 (2007), 655-682. doi: 10.1137/050638424.
[11]	A. Y. Khapalov, Geometric aspects of force controllability for a swimming model, Appl. Math. Optim., 57 (2008), 98-124. 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), 175-200. 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), 497-513. 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), 790-828. 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), 921-951. 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), 5398-5443. 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), 405-424.
[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), 429-455. doi: 10.1007/s00205-007-0092-2.
[19]	E. D. Sontag, Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition, Springer-Verlag, New-York, 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), 1499-1532.
[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), 53-77. doi: 10.1007/s00021-003-0083-4.
[22]	R. Temam, Problèmes Mathématiques en Plasticité, Gauthier-Villars, Montrouge, 1983.
[23]	J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

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