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 & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
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show all references

References:
[1]

Phys. Rev. Lett., 31 (1973), 125-127.  Google Scholar

[2]

Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4.  Google Scholar

[3]

Comm. Pure Appl. Math., 32 (1979), 555-587. doi: 10.1002/cpa.3160320405.  Google Scholar

[4]

Adv. Diff. Eq., 8 (2003), 443-469.  Google Scholar

[5]

Comm. Math. Phys., 143 (1992), 287-313.  Google Scholar

[6]

C. R. Acad. Sci. Paris, 72 (1871), 755-759. Google Scholar

[7]

J. Nonlinear Science, 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.  Google Scholar

[8]

Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[9]

Appl. Math. Optim., 65 (2012), 221-251. doi: 10.1007/s00245-011-9156-7.  Google Scholar

[10]

Commun. Contemp. Math., 13 (2011), 183-189. doi: 10.1142/S021919971100418X.  Google Scholar

[11]

ESAIM Control Optim. Calc. Var., 8 (2002), 513-554. doi: 10.1051/cocv:2002050.  Google Scholar

[12]

Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.  Google Scholar

[13]

Ph.D thesis, Federal University of Rio de Janeiro, 1994. Google Scholar

[14]

SIAM J. Control Optimization, 15 (1977), 185-220.  Google Scholar

[15]

in "European Control Conference,'' Karlsruhe, Germany, 1999. Google Scholar

[16]

Stud. in Appl. Math., 70 (1984), 235-258.  Google Scholar

[17]

Math. Z., 41 (1936), 367-369. doi: 10.1007/BF01180426.  Google Scholar

[18]

Comm. Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336.  Google Scholar

[19]

Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417.  Google Scholar

[20]

J. Differential Equations, 246 (2009), 1342-1353. doi: 10.1016/j.jde.2008.11.002.  Google Scholar

[21]

Proc. Amer. Math. Soc., 135 (2007), 1515-1522. 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]

Tome 1, Masson, Paris, 1988. Google Scholar

[24]

Differential Integral Equations, 22 (2009), 53-68.  Google Scholar

[25]

Quart. Appl. Math., 69 (2011), 723-746. doi: 10.1090/S0033-569X-2011-01245-6.  Google Scholar

[26]

Math. Methods Appl. Sci., 30 (2007), 1419-1435. doi: 10.1002/mma.847.  Google Scholar

[27]

Quart. Appl. Math., 60 (2002), 111-129.  Google Scholar

[28]

in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024.  Google Scholar

[29]

Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600.  Google Scholar

[30]

Discrete Contin. Dyn. Syst., 24 (2009), 273-313. doi: 10.3934/dcds.2009.24.273.  Google Scholar

[31]

Math. Control Relat. Fields, 1 (2011), 353-389. doi: 10.3934/mcrf.2011.1.353.  Google Scholar

[32]

ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015.  Google Scholar

[33]

Systems Control Lett., 57 (2008), 595-601. doi: 10.1016/j.sysconle.2007.12.009.  Google Scholar

[34]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1511-1535. 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]

IEEE Trans. Automat. Control, 47 (2002), 594-609. doi: 10.1109/9.995037.  Google Scholar

[37]

Systems Control Lett., 52 (2004), 167-178. doi: 10.1016/j.sysconle.2003.11.008.  Google Scholar

[38]

ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102.  Google Scholar

[39]

SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229.  Google Scholar

[40]

Comput. Appl. Math., 21 (2002), 355-367.  Google Scholar

[41]

ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012.  Google Scholar

[42]

SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.  Google Scholar

[43]

J. Syst. Sci. Complex., 22 (2009), 647-682. doi: 10.1007/s11424-009-9194-2.  Google Scholar

[44]

J. Differential Equations, 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014.  Google Scholar

[45]

M2AN Math. Model. Numer. Anal., 34 (2000), 501-523. doi: 10.1051/m2an:2000153.  Google Scholar

[46]

Ph.D thesis, Federal University of Rio de Janeiro, 2001. Google Scholar

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