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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  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 & Control Theory, doi: 10.3934/eect.2020029
References:
[1]

C. BaiocchiV. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956.  Google Scholar

[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.  Google Scholar

[3]

J. L. BonaG. PonceJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.   Google Scholar

[15]

A. Haraux, Quelques Mhodes et Résultats Récents en Théorie de la Contrôlabilité Exacte, Rapport de recherche No. 1317, INRIA Rocquencourt, Octobre 1990. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[21]

S. JaffardM. Tucsnak and E. Zuazua, On a theorem of Ingham, J. Fourier Anal. Appl., 3 (1997), 577-582.  doi: 10.1007/BF02648885.  Google Scholar

[22]

V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.  doi: 10.1137/S0363012996301609.  Google Scholar

[23]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar

[24]

V. KomornikD. 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.   Google Scholar

[25]

J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Volume I, Masson, Paris, 1988.  Google Scholar

[26]

P. Loreti, On some gap theorems, Proceedings of the 11th Meeting of EWM, CWI Tract (Marseille, France), 135 (2005), 39-45.  Google Scholar

[27]

M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258.   Google Scholar

[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.  Google Scholar

[29]

S. MicuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

C. BaiocchiV. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956.  Google Scholar

[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.  Google Scholar

[3]

J. L. BonaG. PonceJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.   Google Scholar

[15]

A. Haraux, Quelques Mhodes et Résultats Récents en Théorie de la Contrôlabilité Exacte, Rapport de recherche No. 1317, INRIA Rocquencourt, Octobre 1990. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[21]

S. JaffardM. Tucsnak and E. Zuazua, On a theorem of Ingham, J. Fourier Anal. Appl., 3 (1997), 577-582.  doi: 10.1007/BF02648885.  Google Scholar

[22]

V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.  doi: 10.1137/S0363012996301609.  Google Scholar

[23]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.  Google Scholar

[24]

V. KomornikD. 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.   Google Scholar

[25]

J.-L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Volume I, Masson, Paris, 1988.  Google Scholar

[26]

P. Loreti, On some gap theorems, Proceedings of the 11th Meeting of EWM, CWI Tract (Marseille, France), 135 (2005), 39-45.  Google Scholar

[27]

M. Mehrenberger, Critical length for a Beurling type theorem, Bol. Un. Mat. Ital. B, 8 (2005), 251-258.   Google Scholar

[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.  Google Scholar

[29]

S. MicuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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