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doi: 10.3934/cpaa.2021158
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## Controllability and stabilization of gravity-capillary surface water waves in a basin

 1 Huawei Beijing Research Center, 156 Beiqing Rd, Haidian District, Beijing, 100039, China 2 Beijing National Day School, No. 66 Yuquan Lu, Haidian District, Beijing, 100039, China 3 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

* Corresponding author

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

Received  January 2021 Revised  July 2021 Early access September 2021

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.

Citation: Jing Cui, Guangyue Gao, Shu-Ming Sun. Controllability and stabilization of gravity-capillary surface water waves in a basin. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021158
##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] S. Avdonin and S. Ivanov, Families of Exponentials, Cambridge University Press, Cambridge, UK, 1995.   Google Scholar [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.  Google Scholar [6] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16 (1978) 373–379 doi: 10.1137/0316023.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [10] P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman, Boston, 1985.  Google Scholar [11] L. M. Hocking, Capillary-gravity waves produced by a heaving body, J. Fluid Mech., 186 (1986), 337-349.   Google Scholar [12] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Zeit., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972.  Google Scholar [17] J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [21] S. Mottelet, Controllability and stabilization of a canal with wave generators, SIAM J. Control Optim., 38 (2000), 711-735.  doi: 10.1137/S0363012998347134.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [28] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.  Google Scholar [29] H. Zhu, Control of three dimensional water waves, Arch. Ration. Mech. Anal., 236 (2020), 893-966.  doi: 10.1007/s00205-019-01485-3.  Google Scholar

show all references

##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] S. Avdonin and S. Ivanov, Families of Exponentials, Cambridge University Press, Cambridge, UK, 1995.   Google Scholar [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.  Google Scholar [6] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16 (1978) 373–379 doi: 10.1137/0316023.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [10] P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman, Boston, 1985.  Google Scholar [11] L. M. Hocking, Capillary-gravity waves produced by a heaving body, J. Fluid Mech., 186 (1986), 337-349.   Google Scholar [12] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Zeit., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, New York, 1972.  Google Scholar [17] J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [21] S. Mottelet, Controllability and stabilization of a canal with wave generators, SIAM J. Control Optim., 38 (2000), 711-735.  doi: 10.1137/S0363012998347134.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [28] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.  Google Scholar [29] H. Zhu, Control of three dimensional water waves, Arch. Ration. Mech. Anal., 236 (2020), 893-966.  doi: 10.1007/s00205-019-01485-3.  Google Scholar
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