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June  2022, 21(6): 2035-2063. doi: 10.3934/cpaa.2021158

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 Published  June 2022 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 and Applied Analysis, 2022, 21 (6) : 2035-2063. 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.

[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. AlazardP. 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. ShenS. 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.

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.

[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. AlazardP. 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. ShenS. 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.

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