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Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.
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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 |
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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