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


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  • [1]

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


    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.


    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.


    F. Bonnans, Optimisation Continue, Dunod, Paris, 2006.


    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.


    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.


    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.


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


    S. Court, Existence of 3D Strong Solutions for a System Modeling a Deformable Solid in a Viscous Incompressible Fluid, arXiv:1303.0163.


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


    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.


    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).


    A. Y. Khapalov, Local controllability for a "swimming'' model, SIAM J. Control Optim., 46 (2007), 655-682.doi: 10.1137/050638424.


    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.


    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.


    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.


    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.


    J. P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.doi: 10.1137/080744761.


    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.


    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.


    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.

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