# American Institute of Mathematical Sciences

June  2013, 2(2): 379-402. doi: 10.3934/eect.2013.2.379

## Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system

 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil 2 Institut Elie Cartan de Lorraine, UMR 7502 UdL/CNRS/INRIA, B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France

Received  September 2012 Revised  February 2013 Published  March 2013

This paper is concerned with the exact controllability problem for a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to one-dimensional horizontal motion. We take as fluid model a Boussinesq system of KdV-KdV type, and as control the acceleration of the tank. We derive for the linearized system an exact controllability result in small time in an appropriate space.
Citation: Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations and Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
##### References:
 [1] M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett., 31 (1973), 125-127. [2] E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4. [3] J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Comm. Pure Appl. Math., 32 (1979), 555-587. doi: 10.1002/cpa.3160320405. [4] E. Bisognin, V. Bisognin and G. Perla Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries equations with localized damping, Adv. Diff. Eq., 8 (2003), 443-469. [5] J. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. [6] J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Acad. Sci. Paris, 72 (1871), 755-759. [7] J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Science, 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4. [8] J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory, Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010. [9] M. M. Cavalcanti, V. N. Domingos Cavalcanti, A. Faminskii and F. Natali, Decay of solutions to damped Korteweg-de Vries type equation, Appl. Math. Optim., 65 (2012), 221-251. doi: 10.1007/s00245-011-9156-7. [10] E. Cerpa and A. F. Pazoto, A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations'' [MR2561938], Commun. Contemp. Math., 13 (2011), 183-189. doi: 10.1142/S021919971100418X. [11] J.-M. Coron, Local Controllability of a 1-D tank containing a fluid modeled by the shallow water equations, ESAIM Control Optim. Calc. Var., 8 (2002), 513-554. doi: 10.1051/cocv:2002050. [12] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563. [13] M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations," Ph.D thesis, Federal University of Rio de Janeiro, 1994. [14] S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optimization, 15 (1977), 185-220. [15] F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, in "European Control Conference,'' Karlsruhe, Germany, 1999. [16] J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves, Stud. in Appl. Math., 70 (1984), 235-258. [17] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-369. doi: 10.1007/BF01180426. [18] C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336. [19] F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417. [20] F. Linares and A. F. Pazoto, Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane, J. Differential Equations, 246 (2009), 1342-1353. doi: 10.1016/j.jde.2008.11.002. [21] F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7. [22] F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain,, Trans. Amer. Math. Soc., (). [23] J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués," Tome 1, Masson, Paris, 1988. [24] C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equation as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations, 22 (2009), 53-68. [25] G. P. Menzala, C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a class of KdV equations with localized damping, Quart. Appl. Math., 69 (2011), 723-746. doi: 10.1090/S0033-569X-2011-01245-6. [26] C. P. Massarolo, G. P. Menzala and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping, Math. Methods Appl. Sci., 30 (2007), 1419-1435. doi: 10.1002/mma.847. [27] G. P. Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129. [28] S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024. [29] S. Micu, J. H. Ortega and A. F. Pazoto, On the controllability of a nonlinear coupled system of Korteweg-de Vries equations, Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600. [30] S. Micu, J. H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313. doi: 10.3934/dcds.2009.24.273. [31] D. Nina, A. F. Pazoto and L. Rosier, Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain, Math. Control Relat. Fields, 1 (2011), 353-389. doi: 10.3934/mcrf.2011.1.353. [32] A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015. [33] A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems Control Lett., 57 (2008), 595-601. doi: 10.1016/j.sysconle.2007.12.009. [34] A. F. Pazoto and L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1511-1535. doi: 10.3934/dcdsb.2010.14.1511. [35] A. F. Pazoto and G. R. Souza, Uniform stabilization of a nonlinear dispersive system,, Quart. Appl. Math., (). [36] N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems, IEEE Trans. Automat. Control, 47 (2002), 594-609. doi: 10.1109/9.995037. [37] C. Prieur and J. de Halleux, Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations, Systems Control Lett., 52 (2004), 167-178. doi: 10.1016/j.sysconle.2003.11.008. [38] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102. [39] L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229. [40] L. Rosier, A fundamental solution supported in a strip for a dispersive equation. Special issue in memory of Jacques-Louis Lions, Comput. Appl. Math., 21 (2002), 355-367. [41] L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012. [42] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409. [43] L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: Recent progresses, J. Syst. Sci. Complex., 22 (2009), 647-682. doi: 10.1007/s11424-009-9194-2. [44] L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations, 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014. [45] J.-C. Saut and N. Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal., 34 (2000), 501-523. doi: 10.1051/m2an:2000153. [46] O. P. Vera Villagran, "Gain of Regularity of the Solutions of a Coupled System of Equations of Korteweg-de Vries Type," Ph.D thesis, Federal University of Rio de Janeiro, 2001.

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##### References:
 [1] M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett., 31 (1973), 125-127. [2] E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4. [3] J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Comm. Pure Appl. Math., 32 (1979), 555-587. doi: 10.1002/cpa.3160320405. [4] E. Bisognin, V. Bisognin and G. Perla Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries equations with localized damping, Adv. Diff. Eq., 8 (2003), 443-469. [5] J. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. [6] J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Acad. Sci. Paris, 72 (1871), 755-759. [7] J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Science, 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4. [8] J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory, Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010. [9] M. M. Cavalcanti, V. N. Domingos Cavalcanti, A. Faminskii and F. Natali, Decay of solutions to damped Korteweg-de Vries type equation, Appl. Math. Optim., 65 (2012), 221-251. doi: 10.1007/s00245-011-9156-7. [10] E. Cerpa and A. F. Pazoto, A note on the paper "On the controllability of a coupled system of two Korteweg-de Vries equations'' [MR2561938], Commun. Contemp. Math., 13 (2011), 183-189. doi: 10.1142/S021919971100418X. [11] J.-M. Coron, Local Controllability of a 1-D tank containing a fluid modeled by the shallow water equations, ESAIM Control Optim. Calc. Var., 8 (2002), 513-554. doi: 10.1051/cocv:2002050. [12] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563. [13] M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations," Ph.D thesis, Federal University of Rio de Janeiro, 1994. [14] S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optimization, 15 (1977), 185-220. [15] F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, in "European Control Conference,'' Karlsruhe, Germany, 1999. [16] J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves, Stud. in Appl. Math., 70 (1984), 235-258. [17] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-369. doi: 10.1007/BF01180426. [18] C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336. [19] F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417. [20] F. Linares and A. F. Pazoto, Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane, J. Differential Equations, 246 (2009), 1342-1353. doi: 10.1016/j.jde.2008.11.002. [21] F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7. [22] F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain,, Trans. Amer. Math. Soc., (). [23] J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués," Tome 1, Masson, Paris, 1988. [24] C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equation as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations, 22 (2009), 53-68. [25] G. P. Menzala, C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a class of KdV equations with localized damping, Quart. Appl. Math., 69 (2011), 723-746. doi: 10.1090/S0033-569X-2011-01245-6. [26] C. P. Massarolo, G. P. Menzala and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping, Math. Methods Appl. Sci., 30 (2007), 1419-1435. doi: 10.1002/mma.847. [27] G. P. Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129. [28] S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024. [29] S. Micu, J. H. Ortega and A. F. Pazoto, On the controllability of a nonlinear coupled system of Korteweg-de Vries equations, Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600. [30] S. Micu, J. H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313. doi: 10.3934/dcds.2009.24.273. [31] D. Nina, A. F. Pazoto and L. Rosier, Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain, Math. Control Relat. Fields, 1 (2011), 353-389. doi: 10.3934/mcrf.2011.1.353. [32] A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015. [33] A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems Control Lett., 57 (2008), 595-601. doi: 10.1016/j.sysconle.2007.12.009. [34] A. F. Pazoto and L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1511-1535. doi: 10.3934/dcdsb.2010.14.1511. [35] A. F. Pazoto and G. R. Souza, Uniform stabilization of a nonlinear dispersive system,, Quart. Appl. Math., (). [36] N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems, IEEE Trans. Automat. Control, 47 (2002), 594-609. doi: 10.1109/9.995037. [37] C. Prieur and J. de Halleux, Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations, Systems Control Lett., 52 (2004), 167-178. doi: 10.1016/j.sysconle.2003.11.008. [38] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102. [39] L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229. [40] L. Rosier, A fundamental solution supported in a strip for a dispersive equation. Special issue in memory of Jacques-Louis Lions, Comput. Appl. Math., 21 (2002), 355-367. [41] L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012. [42] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409. [43] L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: Recent progresses, J. Syst. Sci. Complex., 22 (2009), 647-682. doi: 10.1007/s11424-009-9194-2. [44] L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations, 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014. [45] J.-C. Saut and N. Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal., 34 (2000), 501-523. doi: 10.1051/m2an:2000153. [46] O. P. Vera Villagran, "Gain of Regularity of the Solutions of a Coupled System of Equations of Korteweg-de Vries Type," Ph.D thesis, Federal University of Rio de Janeiro, 2001.
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