Shallow water waves generated by a floating object: A control theoretical perspective

Pei Su; Marius Tucsnak

doi:10.3934/mcrf.2022048

Article Contents

2023, Volume 13, Issue 4: 1529-1555. Doi: 10.3934/mcrf.2022048

This issue Previous Article Strong stationarity for a highly nonsmooth optimization problem with control constraints Next Article Sparse optimal control of a quasilinear elliptic PDE in measure spaces

Shallow water waves generated by a floating object: A control theoretical perspective

Pei Su^, and
Marius Tucsnak

Institut de Mathématiques de Bordeaux (IMB), Université de Bordeaux, 351, Cours de la Libération - F 33 405, France

^*Corresponding author: Pei Su
^*Corresponding author: Pei Su

Received: August 2021

Revised: September 2022

Early access: October 15, 2022

Published online: October 15, 2022

The authors are members of the ETN network ConFlex, funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 765579. The second author acknowledges the support of the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately).

Keywords:

Mathematics Subject Classification: Primary: 93B03, 76B15; Secondary: 35Q35, 93D15.

Citation:

Full Text(HTML)

Figure 1. Floating body in a tank filled with water

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114.
[2]	C. J. K. Batty and Q. P. V u, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc., 322 (1990), 805-818. doi: 10.1090/S0002-9947-1990-1022866-5.
[3]	G. Beck and D. Lannes, Freely floating objects on a fluid governed by the Boussinesq equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39 (2022), 575-646. doi: 10.4171/aihpc/15.
[4]	C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM Journal on Control and Optimization, 16 (1978), 373-379. doi: 10.1137/0316023.
[5]	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.
[6]	D. Bresch, D. Lannes and G. Métivier, Waves interacting with a partially immersed obstacle in the Boussinesq regime, Anal. PDE, 14 (2021), 1085-1124. doi: 10.2140/apde.2021.14.1085.
[7]	N. Cîndea, S. Micu, I. Rovenţa and M. Tucsnak, Particle supported control of a fluid–particle system, Journal de Mathématiques Pures et Appliquées, 104 (2015), 311-353. doi: 10.1016/j.matpur.2015.02.009.
[8]	J. A. Cretel, G. Lightbody, G. P. Thomas and A. W. Lewis, Maximisation of energy capture by a wave-energy point absorber using model predictive control, IFAC Proceedings Volumes, 44 (2011), 3714-3721.
[9]	R. F. Curtain and H. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.
[10]	J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact feedback, SIAM Journal on Control and Optimization, 18 (1980), 311-316. doi: 10.1137/0318022.
[11]	A. Gil, J. Segura and N. M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898717822.
[12]	O. Glass, J. J. Kolumbán and F. Sueur, External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid, Anal. PDE, 13 (2020), 651-684. doi: 10.2140/apde.2020.13.651.
[13]	O. Glass, C. Lacave and F. Sueur, On the motion of a small light body immersed in a two dimensional incompressible perfect fluid with vorticity, Communications in Mathematical Physics, 341 (2016), 1015-1065. doi: 10.1007/s00220-015-2489-3.
[14]	E. Godlewski, M. Parisot, J. Sainte-Marie and F. Wahl, Congested shallow water model: Roof modeling in free surface flow, ESAIM Math. Model. Numer. Anal., 52 (2018), 1679-1707. doi: 10.1051/m2an/2018032.
[15]	F. Guo, K. Guo and C. Zhang, Lack of uniformly exponential stabilization for isometric $C_0$-semigroups under compact perturbation of the generators in Banach spaces, Proceedings of the American Mathematical Society, 135 (2007), 1881-1887. doi: 10.1090/S0002-9939-07-08698-4.
[16]	A. Haraux, Une remarque sur la stabilisation de certains systemes du deuxieme ordre en temps, Portugaliae Mathematica, 46 (1989), 245-258.
[17]	T. Iguchi and D. Lannes, Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana University Mathematics Journal, 70 (2021), 353-364. doi: 10.1512/iumj.2021.70.8201.
[18]	F. John, On the motion of floating bodies. I, Communications on Pure and Applied Mathematics, 2 (1949), 13-57. doi: 10.1002/cpa.3160020102.
[19]	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.
[20]	V. Komornik and P. Loreti, Fourier Series in Control Theory, Monographs in Mathematics, Springer-Verlag, New York, 2005.
[21]	C. Lacave and T. Takahashi, Small moving rigid body into a viscous incompressible fluid, Archive for Rational Mechanics and Analysis, 223 (2017), 1307-1335. doi: 10.1007/s00205-016-1058-z.
[22]	D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.
[23]	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.
[24]	D. Lannes, Modeling shallow water waves, Nonlinearity, 33 (2020), R1-R57. doi: 10.1088/1361-6544/ab6c7c.
[25]	G. Li, G. Weiss, M. Mueller, S. Townley and M. R. Belmont, Wave energy converter control by wave prediction and dynamic programming, Renewable Energy, 48 (2012), 392-403.
[26]	K. Liu, Locally distributed control and damping for the conservative systems, SIAM Journal on Control and Optimization, 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928.
[27]	D. Maity, J. San Martín, T. 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. doi: 10.1007/s00332-019-09536-5.
[28]	N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems, IEEE Transactions on Automatic Control, 47 (2002), 594-609. doi: 10.1109/9.995037.
[29]	A. Roy and T. Takahashi, Stabilization of a rigid body moving in a compressible viscous fluid, Journal of Evolution Equations, 21 (2021), 167-200. doi: 10.1007/s00028-020-00574-1.
[30]	K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation, Pure and Applied Mathematics, No. 26. Marcel Dekker, Inc., New York, 1974.
[31]	P. Su, Asymptotic behaviour of a linearized water waves system in a rectangle, Asymptotic Analysis, Pre-press, (2021), 1-26. doi: 10.3233/ASY-221767.
[32]	P. Su, Stabilization of the Systems Describing Water Waves and Their Interaction with a Floating Object, Ph.D thesis, Université de Bordeaux, 2021.
[33]	P. Su, Strong stabilization of a linearized gravity-capillary water waves system in a tank, 2021 60th IEEE Conference on Decision and Control (CDC), (2021), 6194-6199. doi: 10.1109/CDC45484.2021.9683269.
[34]	P. Su, M. Tucsnak and G. Weiss, Stabilizability properties of a linearized water waves system, Systems Control Lett., 139 (2020), 104672, 10 pp. doi: 10.1016/j.sysconle.2020.104672.
[35]	P. Su, M. Tucsnak and G. Weiss, Strong stabilization of small water waves in a pool, IFAC-PapersOnLine, 54 (2021), 378-383.
[36]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.
[37]	G. Weiss, Admissibility of unbounded control operators, SIAM Journal on Control and Optimization, 27 (1989), 527-545. doi: 10.1137/0327028.
[38]	G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.