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

Sébastien Court

doi:10.3934/eect.2014.3.83

Article Contents

2014, Volume 3, Issue 1: 83-118. Doi: 10.3934/eect.2014.3.83

This issue Previous Article Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system. Next Article Recovery of time dependent volatility coefficient by linearization

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

Sébastien Court^1,

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

Received: February 28, 2013

Revised: November 30, 2013

Published: February 2014

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 93C20, 35Q30, 76D05, 76D07, 74F10, 93C05, 93B52, 93D15; Secondary: 74A99, 35Q74.

Citation:

Related Papers

Cited by

References

[1]	G. Allaire, Conception Optimale de Structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2007.
[2]	G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eq., 9 (2009), 341-370.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, Boston, Cambridge, MA, 1992.
[4]	F. Bonnans, Optimisation Continue, Dunod, Paris, 2006.
[5]	J. P. Bourguignon and H. Brezis, Remarks on the euler equation, J. Funct. Analysis, 15 (1974), 341-363.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), 325-385.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), 2814-2835.doi: 10.1137/110828654.
[8]	P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.
[9]	S. Court, Existence of 3D Strong Solutions for a System Modeling a Deformable Solid in a Viscous Incompressible Fluid, arXiv:1303.0163.
[10]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Springer-Verlag, New York, 1994.
[11]	O. Glass and L. Rosier, On the Control of the Motion of a Boat, Mathematical Models and Methods in Applied Sciences, Volume 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), 217-290 (1992).
[13]	A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Control Optim., 46 (2007), 655-682.doi: 10.1137/050638424.
[14]	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.
[15]	J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers, Acta Appl. Math., 123 (2013), 175-200.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), 790-828.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), 921-951.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), 5398-5443.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), 405-424.
[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), 429-455.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), 1499-1532.