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March  2014, 4(1): 45-99. doi: 10.3934/mcrf.2014.4.45

Control of a Korteweg-de Vries equation: A tutorial

Eduardo Cerpa 1,

1. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso, Chile

Received  January 2012 Revised  January 2013 Published  December 2013

These notes are intended to be a tutorial material revisiting in an almost self-contained way, some control results for the Korteweg-de Vries (KdV) equation posed on a bounded interval. We address the topics of boundary controllability and internal stabilization for this nonlinear control system. Concerning controllability, homogeneous Dirichlet boundary conditions are considered and a control is put on the Neumann boundary condition at the right end-point of the interval. We show the existence of some critical domains for which the linear KdV equation is not controllable. In despite of that, we prove that in these cases the nonlinearity gives the exact controllability. Regarding stabilization, we study the problem where all the boundary conditions are homogeneous. We add an internal damping mechanism in order to force the solutions of the KdV equation to decay exponentially to the origin in $L^2$-norm.
Keywords: nonlinear control., controllability, critical domain, stabilization, Korteweg-de Vries equation.
Mathematics Subject Classification: Primary: 93C20, 35Q53; Secondary: 93B05, 93D1.
Citation: Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
References:
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show all references

References:
[1]

SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[2]

L. Baudouin, E. Cerpa, E. Crépeau and A. Mercado, On the determination of the principal coefficient from boundary measurements in a KdV equation,, J. Inverse Ill-Posed Probl., ().  doi: 10.1515/jip-2013-0015.  Google Scholar

[3]

J. Math. Pures Appl. (9), 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005.  Google Scholar

[4]

J. Funct. Anal., 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021.  Google Scholar

[5]

SIAM J. Math. Anal., 14 (1983), 1056-1106. doi: 10.1137/0514085.  Google Scholar

[6]

Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.  Google Scholar

[7]

SIAM J. Control Optim., 46 (2007), 877-899. doi: 10.1137/06065369X.  Google Scholar

[8]

IEEE Trans. Automat. Control, 58 (2013), 1688-1695. doi: 10.1109/TAC.2013.2241479.  Google Scholar

[9]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457-475. doi: 10.1016/j.anihpc.2007.11.003.  Google Scholar

[10]

Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 655-668. doi: 10.3934/dcdsb.2009.11.655.  Google Scholar

[11]

SIAM J. Control Optim., 51 (2013), 2976-3010. doi: 10.1137/120891721.  Google Scholar

[12]

Comm. in Contemp. Math., 11 (2009), 495-521. doi: 10.1142/S0219199709003454.  Google Scholar

[13]

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

[14]

J. Math. Pures Appl. (9), 75 (1996), 155-188.  Google Scholar

[15]

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

[16]

C. R. Acad. Sci. Paris, 342 (2006), 103-108. doi: 10.1016/j.crma.2005.11.004.  Google Scholar

[17]

Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[18]

J. Eur. Math. Soc., 6 (2004), 367-398.  Google Scholar

[19]

SIAM J. Control Optim., 43 (2004), 549-569. doi: 10.1137/S036301290342471X.  Google Scholar

[20]

Internat. J. Control, 74 (2001), 1096-1106. doi: 10.1080/00207170110052202.  Google Scholar

[21]

Ph.D thesis, Université Paris-Sud, Paris, 2002. Google Scholar

[22]

J. Funct. Anal., 244 (2007), 504-535. doi: 10.1016/j.jfa.2006.11.004.  Google Scholar

[23]

Differential Integral Equations, 20 (2007), 601-642.  Google Scholar

[24]

Asymptot. Anal., 60 (2008), 61-100.  Google Scholar

[25]

Systems Control Lett., 59 (2010), 390-395. doi: 10.1016/j.sysconle.2010.05.001.  Google Scholar

[26]

Math. Res. Lett., 10 (2003), 833-846. doi: 10.4310/MRL.2003.v10.n6.a10.  Google Scholar

[27]

Philos. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.  Google Scholar

[28]

Advances in Design and Control, 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.  Google Scholar

[29]

Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.  Google Scholar

[30]

Universitext, Springer, New York, 2009.  Google Scholar

[31]

Recherches en Mathématiques Appliquées, 9, Masson, Paris, 1988.  Google Scholar

[32]

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

[33]

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

[34]

Applied Mathematical Sciences, 44, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar

[41]

SIAM J. Control Optim., 31 (1993), 659-676. doi: 10.1137/0331030.  Google Scholar

[42]

J. Math. Anal. Appl., 190 (1995), 449-488. doi: 10.1006/jmaa.1995.1087.  Google Scholar

[43]

J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[44]

Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[45]

SIAM J. Control Optim., 34 (1996), 892-912. doi: 10.1137/S0363012994269491.  Google Scholar

[46]

CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.  Google Scholar

[47]

SIAM J. Control Optim., 43 (2005), 2233-2244. doi: 10.1137/S0363012901388452.  Google Scholar

[48]

Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[49]

SIAM J. Math. Anal., 23 (1992), 55-71. doi: 10.1137/0523004.  Google Scholar

[50]

SIAM J. Control Optim., 37 (1999), 543-565. doi: 10.1137/S0363012997327501.  Google Scholar

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