March  2014, 3(1): 83-118. doi: 10.3934/eect.2014.3.83

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

1. 

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

Received  March 2013 Revised  December 2013 Published  February 2014

In this second part we prove that the full nonlinear fluid-solid system introduced in Part I is stabilizable by deformations of the solid that have to satisfy nonlinear constraints. Some of these constraints are physical and guarantee the self-propelled nature of the solid. The proof is based on the boundary feedback stabilization of the linearized system. From this boundary feedback operator we construct a deformation of the solid which satisfies the aforementioned constraints and which stabilizes the nonlinear system. The proof is made by a fixed point method.
Citation: Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83
References:
[1]

G. Allaire, Conception Optimale de Structures,, Mathématiques & Applications (Berlin) [Mathematics & Applications], (2007).

[2]

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.

[3]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 1,, Birkhäuser, (1992).

[4]

F. Bonnans, Optimisation Continue,, Dunod, (2006).

[5]

J. P. Bourguignon and H. Brezis, Remarks on the euler equation,, J. Funct. Analysis, 15 (1974), 341. doi: 10.1016/0022-1236(74)90027-5.

[6]

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.

[7]

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.

[8]

P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity,, North-Holland, (1988).

[9]

S. Court, Existence of 3D Strong Solutions for a System Modeling a Deformable Solid in a Viscous Incompressible Fluid,, , ().

[10]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I,, Springer-Verlag, (1994).

[11]

O. Glass and L. Rosier, On the Control of the Motion of a Boat,, Mathematical Models and Methods in Applied Sciences, 23 (2013). doi: 10.1142/S0218202512500571.

[12]

G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods,, Math. Scand., 69 (1991), 217.

[13]

A. Y. Khapalov, Local controllability for a "swimming'' model,, SIAM J. Control Optim., 46 (2007), 655. doi: 10.1137/050638424.

[14]

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.

[15]

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.

[16]

J. P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726.

[17]

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.

[18]

J. P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398. doi: 10.1137/080744761.

[19]

J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms,, Quart. Appl. Math., 65 (2007), 405.

[20]

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.

[21]

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.

show all references

References:
[1]

G. Allaire, Conception Optimale de Structures,, Mathématiques & Applications (Berlin) [Mathematics & Applications], (2007).

[2]

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.

[3]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 1,, Birkhäuser, (1992).

[4]

F. Bonnans, Optimisation Continue,, Dunod, (2006).

[5]

J. P. Bourguignon and H. Brezis, Remarks on the euler equation,, J. Funct. Analysis, 15 (1974), 341. doi: 10.1016/0022-1236(74)90027-5.

[6]

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.

[7]

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.

[8]

P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity,, North-Holland, (1988).

[9]

S. Court, Existence of 3D Strong Solutions for a System Modeling a Deformable Solid in a Viscous Incompressible Fluid,, , ().

[10]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I,, Springer-Verlag, (1994).

[11]

O. Glass and L. Rosier, On the Control of the Motion of a Boat,, Mathematical Models and Methods in Applied Sciences, 23 (2013). doi: 10.1142/S0218202512500571.

[12]

G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods,, Math. Scand., 69 (1991), 217.

[13]

A. Y. Khapalov, Local controllability for a "swimming'' model,, SIAM J. Control Optim., 46 (2007), 655. doi: 10.1137/050638424.

[14]

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.

[15]

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.

[16]

J. P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726.

[17]

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.

[18]

J. P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398. doi: 10.1137/080744761.

[19]

J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms,, Quart. Appl. Math., 65 (2007), 405.

[20]

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.

[21]

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.

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