# American Institute of Mathematical Sciences

September  2020, 9(3): 693-719. doi: 10.3934/eect.2020029

## Pointwise control of the linearized Gear-Grimshaw system

 1 Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-545, Brazil 2 College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, China 3 Département de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France 4 Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

* Corresponding author

Received  June 2019 Revised  September 2019 Published  September 2020 Early access  December 2019

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability by using two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.

Citation: Roberto de A. Capistrano-Filho, Vilmos Komornik, Ademir F. Pazoto. Pointwise control of the linearized Gear-Grimshaw system. Evolution Equations and Control Theory, 2020, 9 (3) : 693-719. doi: 10.3934/eect.2020029
##### References:
 [1] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956. [2] A. Beurling, Interpolation for an Interval in, $\mathbb{R}^1$ in The Collected Works of Arne Beurling. Vol. 2. Harmonic Analysis (eds. L. Carleson, P. Malliavin, J. Neuberger and J. Wermer), Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989. [3] J. L. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Physics, 143 (1992), 287-313.  doi: 10.1007/BF02099010. [4] R. A. Capistrano-Filho, A. F. Pazoto and V. Komornik, Stabilization of the Gear-Grimshaw system on a periodic domain, Commun. Contemp. Math., 16 (2014), 1450047, 22 pp. doi: 10.1142/S0219199714500473. [5] R. A. Capistrano-Filho and F. A. Gallego, Asymptotic behavior of Boussinesq system of KdV-KdV type, J. Differential Equations, 265 (2018), 2341-2374.  doi: 10.1016/j.jde.2018.04.034. [6] R. A. Capistrano-Filho, F. A. Gallego and A. F. Pazoto, Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictionson on the spatial domain, Z. Angew. Math. Phys., 67 (2016), Art. 109, 36 pp. doi: 10.1007/s00033-016-0705-4. [7] R. A. Capistrano-Filho, F. A. Gallego and A. F. Pazoto, Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain, Math. Control Signals Systems, 29 (2017), Art. 6, 37 pp. doi: 10.1007/s00498-017-0186-9. [8] C. Castro and E. Zuazua, Une remarque sur les séries de Fourier non-harmoniques et son application à la contrôlabilité des cordes avec densité singulière, C. R. Acad. Sci. Paris Sŕ. I, 323 (1996), 365-370. [9] E. Cerpa and A. F. Pazoto, A note on the paper On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math., 13 (2011), 183-189.  doi: 10.1142/S021919971100418X. [10] M. Dávila, Estabilização de um sistema acoplado de equaç oes tipo KdV, Proceedings of the $45^\circ$ Seminário Brasileiro de Análise, Vol I (Florianópolis, Brazil), 1997,453-458. [11] M. Dávila and F. S. Chaves, Infinite conservation laws for a system of Korteweg-de Vries type equations, Proceedings of DINCON 2006, Brazilian Conference on Dynamics, Control and Their Applications (Guaratinguetá, Brazil), 2006, 22-26. [12] S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Opt., 15 (1977), 185-220.  doi: 10.1137/0315015. [13] J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258.  doi: 10.1002/sapm1984703235. [14] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. [15] A. Haraux, Quelques Mhodes et Résultats Récents en Théorie de la Contrôlabilité Exacte, Rapport de recherche No. 1317, INRIA Rocquencourt, Octobre 1990. [16] A. Haraux and S. Jaffard, Pointwise and spectral control of plate vibrations, Rev. Mat. Iberoamericana, 7 (1991), 1-24.  doi: 10.4171/RMI/103. [17] A. Haraux, Quelques propriétés des séries lacunaires utiles dans létude des systèmes élastiques, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, XII (Paris, 1991-1993), 113-124, Pitman Res. Notes Math.Ser., 302, Longman Sci.Tech.Harlow, 1994. [18] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408.  doi: 10.1016/0375-9601(81)90423-0. [19] R. Hirota and J. Satsuma, A Coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Japan, 51 (1982), 3390-3397.  doi: 10.1143/JPSJ.51.3390. [20] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426. [21] S. Jaffard, M. Tucsnak and E. Zuazua, On a theorem of Ingham, J. Fourier Anal. Appl., 3 (1997), 577-582.  doi: 10.1007/BF02648885. [22] V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.  doi: 10.1137/S0363012996301609. [23] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. [24] V. Komornik, D. L. Russell and B.-Y. Zhang, Stabilisation de l'équation de Korteweg-de Vries, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 841-843. [25] J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Volume I, Masson, Paris, 1988. [26] P. Loreti, On some gap theorems, Proceedings of the 11th Meeting of EWM, CWI Tract (Marseille, France), 135 (2005), 39-45. [27] M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258. [28] S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, Spain), SIAM, Philadelphia, PA, 2000, 1020-1024. [29] S. Micu, J. H. Ortega and A. F. Pazoto, On the Controllability of a Coupled system of two Korteweg-de Vries equation, Commun. Contemp. Math., 11 (2009), 779-827.  doi: 10.1142/S0219199709003600. [30] A. F. Pazoto and G. R. Souza, Uniform stabilization of a nonlinear dispersive system, Quart. Appl. Math., 72 (2014), 193-208.  doi: 10.1090/S0033-569X-2013-01343-1. [31] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Cal. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102. [32] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

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##### References:
 [1] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956. [2] A. Beurling, Interpolation for an Interval in, $\mathbb{R}^1$ in The Collected Works of Arne Beurling. Vol. 2. Harmonic Analysis (eds. L. Carleson, P. Malliavin, J. Neuberger and J. Wermer), Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989. [3] J. L. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Physics, 143 (1992), 287-313.  doi: 10.1007/BF02099010. [4] R. A. Capistrano-Filho, A. F. Pazoto and V. Komornik, Stabilization of the Gear-Grimshaw system on a periodic domain, Commun. Contemp. Math., 16 (2014), 1450047, 22 pp. doi: 10.1142/S0219199714500473. [5] R. A. Capistrano-Filho and F. A. Gallego, Asymptotic behavior of Boussinesq system of KdV-KdV type, J. Differential Equations, 265 (2018), 2341-2374.  doi: 10.1016/j.jde.2018.04.034. [6] R. A. Capistrano-Filho, F. A. Gallego and A. F. Pazoto, Neumann boundary controllability of the Gear-Grimshaw system with critical size restrictionson on the spatial domain, Z. Angew. Math. Phys., 67 (2016), Art. 109, 36 pp. doi: 10.1007/s00033-016-0705-4. [7] R. A. Capistrano-Filho, F. A. Gallego and A. F. Pazoto, Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain, Math. Control Signals Systems, 29 (2017), Art. 6, 37 pp. doi: 10.1007/s00498-017-0186-9. [8] C. Castro and E. Zuazua, Une remarque sur les séries de Fourier non-harmoniques et son application à la contrôlabilité des cordes avec densité singulière, C. R. Acad. Sci. Paris Sŕ. I, 323 (1996), 365-370. [9] E. Cerpa and A. F. Pazoto, A note on the paper On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math., 13 (2011), 183-189.  doi: 10.1142/S021919971100418X. [10] M. Dávila, Estabilização de um sistema acoplado de equaç oes tipo KdV, Proceedings of the $45^\circ$ Seminário Brasileiro de Análise, Vol I (Florianópolis, Brazil), 1997,453-458. [11] M. Dávila and F. S. Chaves, Infinite conservation laws for a system of Korteweg-de Vries type equations, Proceedings of DINCON 2006, Brazilian Conference on Dynamics, Control and Their Applications (Guaratinguetá, Brazil), 2006, 22-26. [12] S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Opt., 15 (1977), 185-220.  doi: 10.1137/0315015. [13] J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258.  doi: 10.1002/sapm1984703235. [14] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. [15] A. Haraux, Quelques Mhodes et Résultats Récents en Théorie de la Contrôlabilité Exacte, Rapport de recherche No. 1317, INRIA Rocquencourt, Octobre 1990. [16] A. Haraux and S. Jaffard, Pointwise and spectral control of plate vibrations, Rev. Mat. Iberoamericana, 7 (1991), 1-24.  doi: 10.4171/RMI/103. [17] A. Haraux, Quelques propriétés des séries lacunaires utiles dans létude des systèmes élastiques, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, XII (Paris, 1991-1993), 113-124, Pitman Res. Notes Math.Ser., 302, Longman Sci.Tech.Harlow, 1994. [18] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408.  doi: 10.1016/0375-9601(81)90423-0. [19] R. Hirota and J. Satsuma, A Coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Japan, 51 (1982), 3390-3397.  doi: 10.1143/JPSJ.51.3390. [20] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426. [21] S. Jaffard, M. Tucsnak and E. Zuazua, On a theorem of Ingham, J. Fourier Anal. Appl., 3 (1997), 577-582.  doi: 10.1007/BF02648885. [22] V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.  doi: 10.1137/S0363012996301609. [23] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. [24] V. Komornik, D. L. Russell and B.-Y. Zhang, Stabilisation de l'équation de Korteweg-de Vries, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 841-843. [25] J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Volume I, Masson, Paris, 1988. [26] P. Loreti, On some gap theorems, Proceedings of the 11th Meeting of EWM, CWI Tract (Marseille, France), 135 (2005), 39-45. [27] M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258. [28] S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, Spain), SIAM, Philadelphia, PA, 2000, 1020-1024. [29] S. Micu, J. H. Ortega and A. F. Pazoto, On the Controllability of a Coupled system of two Korteweg-de Vries equation, Commun. Contemp. Math., 11 (2009), 779-827.  doi: 10.1142/S0219199709003600. [30] A. F. Pazoto and G. R. Souza, Uniform stabilization of a nonlinear dispersive system, Quart. Appl. Math., 72 (2014), 193-208.  doi: 10.1090/S0033-569X-2013-01343-1. [31] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Cal. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102. [32] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.
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