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LQR control for a system describing the interaction between a floating solid and the surrounding fluid

Dedicated to Professor Hélène Frankowska for her 70th anniversary

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  • This paper studies an infinite time horizon LQR optimal control problem for a system describing, within a linear approximation, the vertical oscillations of a floating solid, coupled with the motion of the free boundary fluid on which it floats. The fluid flow is described by a viscous version of the linearized Saint-Venant equations (shallow water regime). The major difficulty we face is that the domain occupied by the fluid is unbounded so that the system is not exponentially stable. This raises challenges in proving the wellposedness, requiring the combined use of analytic semigroup theory and an interpolation technique. The main contribution of this paper is that, in spite of the lack of exponential stabilizability, we could define a wellposed LQR problem for which a Riccati-based approach to design feedback controls can be implemented.

    Mathematics Subject Classification: 93C20, 93B52, 35Q35, 49J21.

    Citation:

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  • Figure 1.  The body floating in an unbounded fluid

  • [1] W. Arendt and C. J. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.
    [2] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems Control Found. Appl. Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.
    [3] E. Bocchi, Floating structures in shallow water: Local well-posedness in the axisymmetric case, SIAM Journal on Mathematical Analysis, 52 (2020), 306-339.  doi: 10.1137/18M1174180.
    [4] R. F. Curtain and J. C. Oostveen, Necessary and sufficient conditions for strong stability of distributed parameter systems, Systems & Control Letters, 37 (1999), 11-18.  doi: 10.1016/S0167-6911(98)00109-1.
    [5] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts Appl. Math., 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.
    [6] J. Falnes, Interaction between oscillations and waves, in: Ocean Waves and Oscillating Systems: Linear Interactions Including Wave-Energy Extraction, Cambridge University Press, (2002), 43-57. doi: 10.1017/CBO9780511754630.004.
    [7] B. H. HaakD. MaityT. Takahashi and M. Tucsnak, Mathematical analysis of the motion of a rigid body in a compressible Navier-Stokes-Fourier fluid, Mathematische Nachrichten, 292 (2019), 1972-2017.  doi: 10.1002/mana.201700425.
    [8] F. John, On the motion of floating bodies. I, Communications on Pure and Applied Mathematics, 2 (1949), 13-57.  doi: 10.1002/cpa.3160020102.
    [9] F. John, On the motion of floating bodies. II. Simple harmonic motions, Communications on Pure and Applied Mathematics, 3 (1950), 45-101.  doi: 10.1002/cpa.3160030106.
    [10] U. A. Korde and  J. RingwoodHydrodynamic Control of Wave Energy Devices, Cambridge University Press, 2016.  doi: 10.1017/CBO9781139942072.
    [11] S. G. Krantz, A Panorama of Harmonic Analysis, vol. 27, American Mathematical Soc., 2019. doi: 10.1201/9780429275166-3.
    [12] D. Lannes, On the dynamics of floating structures, Annals of PDE, 3 (2017), Paper No. 11, 81 pp. doi: 10.1007/s40818-017-0029-5.
    [13] A. Lunardi, Interpolation Theory, vol. 16, Springer, 2018. doi: 10.1007/978-88-7642-638-4.
    [14] D. MaityJ. San MartínT. Takahashi and M. Tucsnak, Analysis of a simplified model of rigid structure floating in a viscous fluid, Journal of Nonlinear Science, 29 (2019), 1975-2020. 
    [15] D. Maity and M. Tucsnak, A maximal regularity approach to the analysis of some particulate flows, Particles in Flows, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, (2017), 1-75., doi: 10.1007/978-3-319-60282-0_1.
    [16] J. C. OostveenR. F. Curtain and K. Ito, An approximation theory for strongly stabilizing solutions to the operator $LQ$ Riccati equation, SIAM Journal on Control and Optimization, 38 (2000), 1909-1937.  doi: 10.1137/S0363012998339691.
    [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer Science & Business Media, 1983.
    [18] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Springer Science & Business Media, 2009. doi: 10.1007/978-3-7643-8994-9.
    [19] G. Vergara-HermosillaD. Matignon and M. Tucsnak, Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid, IFAC-PapersOnLine, 53 (2020), 7491-7496.  doi: 10.1016/j.ifacol.2020.12.1311.
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