Article Contents
Article Contents

# Controllability and stabilization of gravity-capillary surface water waves in a basin

• * Corresponding author

It is dedicated to Professor Goong Chen on the occasion of his 70th birthday

• The paper concerns the controllability and stabilization of surface water waves in a two-dimensional rectangular basin under the forces of gravity and surface tension. The surface waves are generated by a wave-maker placed at the left side-boundary and it is physical relevant to see whether the surface waves are controllable or can be stabilized using appropriate motion of the wave-maker. Due to the surface tension, an edge condition must be imposed at the contact point between the free surface and a solid boundary. Two types of wave-makers are considered: "flexible" or "rigid". It is shown that the surface waves are approximately controllable, but not exactly controllable, for both "flexible" and "rigid" wave-makers. In addition, under a static feedback to control a "rigid" wave-maker, the strong stability of feedback control system is obtained.

Correction: The page numbers on each page of the PDF file have been corrected. We apologize for any inconvenience this may cause.

Mathematics Subject Classification: Primary: 76B15, 76B45; Secondary: 93C20.

 Citation:

•  [1] T. Alazard, Stabilization of the water-wave equations with surface tension, Ann. Partial Differ. Equ., 3 (2017), 1-41.  doi: 10.1007/s40818-017-0032-x. [2] T. Alazard, Stabilization of gravity water waves, Journal de Mathèmatiques Pures et Appliquèes, 114 (2018), 51-84.  doi: 10.1016/j.matpur.2017.09.012. [3] T. Alazard, P. Baldi and and D. Han-Kwan, Control of water waves, J. Euro. Math. Soc., 20 (2018), 657-745.  doi: 10.4171/JEMS/775. [4] S. Avdonin and  S. Ivanov,  Families of Exponentials, Cambridge University Press, Cambridge, UK, 1995. [5] K. Balachandran and J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey, J. Optim. Theory Appl., 115 (2002), 7-28.  doi: 10.1023/A:1019668728098. [6] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16 (1978) 373–379 doi: 10.1137/0316023. [7] T. B. Benjamin and F. Ursell, The stability of the plane free surface of a liquid in a vertical periodic motion, Proc. Roy. Soc. Ser. A, 225 (1954), 505-515.  doi: 10.1098/rspa.1954.0218. [8] R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-1-4612-4224-6. [9] D. V. Evans, The effect of surface tension on the waves produced by a heaving circular cylinder, Proc. Cambridge Philos. Soc., 64 (1968), 833-847. [10] P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman, Boston, 1985. [11] L. M. Hocking, Capillary-gravity waves produced by a heaving body, J. Fluid Mech., 186 (1986), 337-349. [12] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Zeit., 41 (1936), 367-379.  doi: 10.1007/BF01180426. [13] G. Joly, S. Mottelet and J. Yvon, Analysis of the control of wave generators in a canal, in Control of Partial Differential Equations and Applications (Laredo, 1994), Marcel Dekker, New York, (1996), 119–134. [14] V. Komornik, A generalization of Ingham's inequality, in Colloq. Math. Soc. $J\grave{a}nos$ Bolyai, Differential Equations Applications, 62 (1991), 213–217. [15] I. Lasiecka and R. Triggiani, Finite rank, relatively bounded perturbations of c-semi-groups, part II: Spectrum allocation and Riesz basis in parabolic and hyperbolic feedback systems, Ann. Mat. Pura Appl., CXLIII (1986), 47-100.  doi: 10.1007/BF01769210. [16] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. [17] J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001. [18] J. L. Lions, Contôlabilité exacte, perturbation et stabilisation de systémes distribués 1, 2, in Collection Recherches en Mathématiques Appliquées, Vol. 8, 9, Masson, Paris, 1988. [19] S. Mottelet, Quelques Aspects Théoriques et Numériques du Contôle d'un Bassin de Carénes, Ph.D. thesis, Université de Technologie de Compiégne, Compiégne, France, 1994. [20] S. Mottelet, G. Joly and J. Yvon, Design of a feedback controller for wave generators in a canal using $H^{\infty}$ methods, in System Modelling and Optimizaation, Lecture Notes in Control and Inform, Springer-Verlag, London, 1994. doi: 10.1007/BFb0035521. [21] S. Mottelet, Controllability and stabilization of a canal with wave generators, SIAM J. Control Optim., 38 (2000), 711-735.  doi: 10.1137/S0363012998347134. [22] M. D. Quinn and N. Carmichael, An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses, Numer. Funct. Anal. Optim., 7 (1984/1985), 197-219.  doi: 10.1080/01630568508816189. [23] P. F. Rhodes-Robinson, On the forced surface waves due to a vertical wave maker in the presence of surface tension, Proc. Cambridge Philos. Soc., 70 (1971), 323-337.  doi: 10.1017/s0305004100049926. [24] M. C. Shen, S. M. Sun and D. Y. Hsieh, Forced capillary-gravity waves in a circular basin, Wave Motion, 18 (1993), 401-412.  doi: 10.1016/0165-2125(93)90068-Q. [25] R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 (1977), 407-411.  doi: 10.1137/0315028. [26] R. Triggiani, Addendum: "A note on the lack of exact controllability for mild solutions in Banach spaces", SIAM J. Control Optim., 18 (1980), 98-99.  doi: 10.1137/0318007. [27] R. Triggiani, Finite rank, relatively bounded perturbations of semi-groups generators, part III: A sharp result on the lack of uniform stabilization, Differ. Integral Equ., 3 (1990), 503-522. [28] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974. [29] H. Zhu, Control of three dimensional water waves, Arch. Ration. Mech. Anal., 236 (2020), 893-966.  doi: 10.1007/s00205-019-01485-3.