Controllability and stabilization of gravity-capillary surface water waves in a basin. (English) Zbl 1487.76025
Summary: 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.
MSC:
| 76B75 | Flow control and optimization for incompressible inviscid fluids |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
| 93C20 | Control/observation systems governed by partial differential equations |
Keywords:
feedback control; stabilization; gravity-capillary wave; rectangular basin; rigid wave-marker; contact edge condition; flexible wave-markerReferences:
| [1] | T. Alazard, Stabilization of the water-wave equations with surface tension, Ann. Partial Differ. Equ., 3, 1-41 (2017) · Zbl 1403.35226 · doi:10.1007/s40818-017-0032-x |
| [2] | T. Alazard, Stabilization of gravity water waves, Journal de Mathèmatiques Pures et Appliquèes, 114, 51-84 (2018) · Zbl 1393.35253 · doi:10.1016/j.matpur.2017.09.012 |
| [3] | T. Alazard; P. Baldi; and D. Han-Kwan, Control of water waves, J. Euro. Math. Soc., 20, 657-745 (2018) · Zbl 1384.93024 · doi:10.4171/JEMS/775 |
| [4] | K. Balachandran; J. P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey, J. Optim. Theory Appl., 115, 7-28 (2002) · Zbl 1023.93010 · doi:10.1023/A:1019668728098 |
| [5] | C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16 (1978) 373-379 · Zbl 0384.93035 |
| [6] | T. B. Benjamin; F. Ursell, The stability of the plane free surface of a liquid in a vertical periodic motion, Proc. Roy. Soc. Ser. A, 225, 505-515 (1954) · Zbl 0057.18801 · doi:10.1098/rspa.1954.0218 |
| [7] | R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1995. · Zbl 0839.93001 |
| [8] | D. V. Evans, The effect of surface tension on the waves produced by a heaving circular cylinder, Proc. Cambridge Philos. Soc., 64, 833-847 (1968) · Zbl 0164.56402 |
| [9] | P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman, Boston, 1985. · Zbl 0695.35060 |
| [10] | L. M. Hocking, Capillary-gravity waves produced by a heaving body, J. Fluid Mech., 186, 337-349 (1986) · Zbl 0644.76019 |
| [11] | A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Zeit., 41, 367-379 (1936) · Zbl 0014.21503 · doi:10.1007/BF01180426 |
| [12] | 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. · Zbl 0845.93065 |
| [13] | V. Komornik, A generalization of Ingham’s inequality, in Colloq. Math. Soc. \(J\grave{a}nos\) Bolyai, Differential Equations Applications, 62 (1991), 213-217. · Zbl 0813.30005 |
| [14] | I. Lasiecka; 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, 47-100 (1986) · Zbl 0623.47040 · doi:10.1007/BF01769210 |
| [15] | J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972. · Zbl 0227.35001 |
| [16] | J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter systems, SIAM Rev., 30, 1-68 (1988) · Zbl 0644.49028 · doi:10.1137/1030001 |
| [17] | 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. · Zbl 0653.93002 |
| [18] | 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. |
| [19] | 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. · Zbl 0825.93577 |
| [20] | S. Mottelet, Controllability and stabilization of a canal with wave generators, SIAM J. Control Optim., 38, 711-735 (2000) · Zbl 0966.76015 · doi:10.1137/S0363012998347134 |
| [21] | M. D. Quinn; N. Carmichael, An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses, Numer. Funct. Anal. Optim., 7, 197-219 (1984/1985) · Zbl 0563.93013 · doi:10.1080/01630568508816189 |
| [22] | 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, 323-337 (1971) · Zbl 0235.76006 · doi:10.1017/s0305004100049926 |
| [23] | M. C. Shen; S. M. Sun; D. Y. Hsieh, Forced capillary-gravity waves in a circular basin, Wave Motion, 18, 401-412 (1993) · Zbl 0804.76015 · doi:10.1016/0165-2125(93)90068-Q |
| [24] | R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15, 407-411 (1977) · Zbl 0354.93014 · doi:10.1137/0315028 |
| [25] | R. Triggiani, Addendum: “A note on the lack of exact controllability for mild solutions in Banach spaces”, SIAM J. Control Optim., 18, 98-99 (1980) · Zbl 0426.93013 · doi:10.1137/0318007 |
| [26] | 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, 503-522 (1990) · Zbl 0773.47021 |
| [27] | G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974. · Zbl 0373.76001 |
| [28] | H. Zhu, Control of three dimensional water waves, Arch. Ration. Mech. Anal., 236, 893-966 (2020) · Zbl 1431.76034 · doi:10.1007/s00205-019-01485-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
