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

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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

Dugan Nina ^1, , Ademir Fernando Pazoto ^1, and Lionel Rosier ^2,

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

Abstract
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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.

Keywords: Korteweg-de Vries equation., Boussinesq system, tank problem, Controllability.

Mathematics Subject Classification: Primary: 93B05, 35Q53; Secondary: 93C2.

Citation: Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & 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. Google Scholar
[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. doi: 10.1016/S0362-546X(97)00724-4. Google Scholar
[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. doi: 10.1002/cpa.3160320405. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal,, C. R. Acad. Sci. Paris, 72 (1871), 755. Google Scholar
[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. doi: 10.1007/s00332-002-0466-4. Google Scholar
[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. doi: 10.1088/0951-7715/17/3/010. Google Scholar
[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. doi: 10.1007/s00245-011-9156-7. Google Scholar
[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. doi: 10.1142/S021919971100418X. Google Scholar
[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. doi: 10.1051/cocv:2002050. Google Scholar
[12]	J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signals Systems, 5 (1992), 295. doi: 10.1007/BF01211563. Google Scholar
[13]	M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations,", Ph.D thesis, (1994). Google Scholar
[14]	S. Dolecki and D. L. Russell, A general theory of observation and control,, SIAM J. Control Optimization, 15 (1977), 185. Google Scholar
[15]	F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid,, in, (1999). Google Scholar
[16]	J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves,, Stud. in Appl. Math., 70 (1984), 235. Google Scholar
[17]	A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar
[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. doi: 10.1080/03605300903585336. Google Scholar
[19]	F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations,, Commun. Pure Appl. Anal., 3 (2004), 417. doi: 10.3934/cpaa.2004.3.417. Google Scholar
[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. doi: 10.1016/j.jde.2008.11.002. Google Scholar
[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. doi: 10.1090/S0002-9939-07-08810-7. Google Scholar
[22]	F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain,, Trans. Amer. Math. Soc., (). Google Scholar
[23]	J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués,", Tome 1, (1988). Google Scholar
[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. Google Scholar
[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. doi: 10.1090/S0033-569X-2011-01245-6. Google Scholar
[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. doi: 10.1002/mma.847. Google Scholar
[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. Google Scholar
[28]	S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations,, in, (2000), 1020. Google Scholar
[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. doi: 10.1142/S0219199709003600. Google Scholar
[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. doi: 10.3934/dcds.2009.24.273. Google Scholar
[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. doi: 10.3934/mcrf.2011.1.353. Google Scholar
[32]	A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping,, ESAIM Control Optim. Calc. Var., 11 (2005), 473. doi: 10.1051/cocv:2005015. Google Scholar
[33]	A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type,, Systems Control Lett., 57 (2008), 595. doi: 10.1016/j.sysconle.2007.12.009. Google Scholar
[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. doi: 10.3934/dcdsb.2010.14.1511. Google Scholar
[35]	A. F. Pazoto and G. R. Souza, Uniform stabilization of a nonlinear dispersive system,, Quart. Appl. Math., (). Google Scholar
[36]	N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems,, IEEE Trans. Automat. Control, 47 (2002), 594. doi: 10.1109/9.995037. Google Scholar
[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. doi: 10.1016/j.sysconle.2003.11.008. Google Scholar
[38]	L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain,, ESAIM Control Optim. Calc. Var., 2 (1997), 33. doi: 10.1051/cocv:1997102. Google Scholar
[39]	L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line,, SIAM J. Control Optim., 39 (2000), 331. doi: 10.1137/S0363012999353229. Google Scholar
[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. Google Scholar
[41]	L. Rosier, Control of the surface of a fluid by a wavemaker,, ESAIM Control Optim. Calc. Var., 10 (2004), 346. doi: 10.1051/cocv:2004012. Google Scholar
[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. doi: 10.1137/050631409. Google Scholar
[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. doi: 10.1007/s11424-009-9194-2. Google Scholar
[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. doi: 10.1016/j.jde.2012.08.014. Google Scholar
[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. doi: 10.1051/m2an:2000153. Google Scholar
[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, (2001). Google Scholar

show all references

References:

[1]	M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical significance,, Phys. Rev. Lett., 31 (1973), 125. Google Scholar
[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. doi: 10.1016/S0362-546X(97)00724-4. Google Scholar
[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. doi: 10.1002/cpa.3160320405. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	J. V. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal,, C. R. Acad. Sci. Paris, 72 (1871), 755. Google Scholar
[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. doi: 10.1007/s00332-002-0466-4. Google Scholar
[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. doi: 10.1088/0951-7715/17/3/010. Google Scholar
[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. doi: 10.1007/s00245-011-9156-7. Google Scholar
[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. doi: 10.1142/S021919971100418X. Google Scholar
[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. doi: 10.1051/cocv:2002050. Google Scholar
[12]	J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signals Systems, 5 (1992), 295. doi: 10.1007/BF01211563. Google Scholar
[13]	M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations,", Ph.D thesis, (1994). Google Scholar
[14]	S. Dolecki and D. L. Russell, A general theory of observation and control,, SIAM J. Control Optimization, 15 (1977), 185. Google Scholar
[15]	F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid,, in, (1999). Google Scholar
[16]	J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves,, Stud. in Appl. Math., 70 (1984), 235. Google Scholar
[17]	A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar
[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. doi: 10.1080/03605300903585336. Google Scholar
[19]	F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations,, Commun. Pure Appl. Anal., 3 (2004), 417. doi: 10.3934/cpaa.2004.3.417. Google Scholar
[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. doi: 10.1016/j.jde.2008.11.002. Google Scholar
[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. doi: 10.1090/S0002-9939-07-08810-7. Google Scholar
[22]	F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain,, Trans. Amer. Math. Soc., (). Google Scholar
[23]	J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués,", Tome 1, (1988). Google Scholar
[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. Google Scholar
[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. doi: 10.1090/S0033-569X-2011-01245-6. Google Scholar
[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. doi: 10.1002/mma.847. Google Scholar
[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. Google Scholar
[28]	S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations,, in, (2000), 1020. Google Scholar
[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. doi: 10.1142/S0219199709003600. Google Scholar
[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. doi: 10.3934/dcds.2009.24.273. Google Scholar
[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. doi: 10.3934/mcrf.2011.1.353. Google Scholar
[32]	A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping,, ESAIM Control Optim. Calc. Var., 11 (2005), 473. doi: 10.1051/cocv:2005015. Google Scholar
[33]	A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type,, Systems Control Lett., 57 (2008), 595. doi: 10.1016/j.sysconle.2007.12.009. Google Scholar
[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. doi: 10.3934/dcdsb.2010.14.1511. Google Scholar
[35]	A. F. Pazoto and G. R. Souza, Uniform stabilization of a nonlinear dispersive system,, Quart. Appl. Math., (). Google Scholar
[36]	N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems,, IEEE Trans. Automat. Control, 47 (2002), 594. doi: 10.1109/9.995037. Google Scholar
[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. doi: 10.1016/j.sysconle.2003.11.008. Google Scholar
[38]	L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain,, ESAIM Control Optim. Calc. Var., 2 (1997), 33. doi: 10.1051/cocv:1997102. Google Scholar
[39]	L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line,, SIAM J. Control Optim., 39 (2000), 331. doi: 10.1137/S0363012999353229. Google Scholar
[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. Google Scholar
[41]	L. Rosier, Control of the surface of a fluid by a wavemaker,, ESAIM Control Optim. Calc. Var., 10 (2004), 346. doi: 10.1051/cocv:2004012. Google Scholar
[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. doi: 10.1137/050631409. Google Scholar
[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. doi: 10.1007/s11424-009-9194-2. Google Scholar
[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. doi: 10.1016/j.jde.2012.08.014. Google Scholar
[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. doi: 10.1051/m2an:2000153. Google Scholar
[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, (2001). Google Scholar

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