×

Global controllability of a nonlinear Korteweg-de Vries equation. (English) Zbl 1170.93006

Summary: We are interested in both the global exact controllability to the trajectories and in the global exact controllability of a nonlinear Korteweg-de Vries equation in a bounded interval. The local exact controllability to the trajectories by means of one boundary control, namely the boundary value at the left endpoint, has already been proved independently by Rosier, and Glass and Guerrero. We first introduce here two more controls: the boundary value at the right endpoint and the right member of the equation, assumed to be \(x\)-independent. Then, we prove that, thanks to these three controls, one has the global exact controllability to the trajectories, for any positive time \(T\). Finally, we introduce a fourth control on the first derivative at the right endpoint, and we get the global exact controllability, for any positive time \(T\).

MSC:

93B05 Controllability
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1081/PDE-120024373 · Zbl 1057.35049 · doi:10.1081/PDE-120024373
[2] DOI: 10.1137/06065369X · Zbl 1147.93005 · doi:10.1137/06065369X
[3] DOI: 10.1016/j.anihpc.2007.11.003 · Zbl 1158.93006 · doi:10.1016/j.anihpc.2007.11.003
[4] DOI: 10.3934/dcdsb.2009.11.655 · Zbl 1161.93018 · doi:10.3934/dcdsb.2009.11.655
[5] DOI: 10.1137/070685749 · Zbl 1282.93050 · doi:10.1137/070685749
[6] DOI: 10.1016/j.crma.2006.12.016 · Zbl 1107.93008 · doi:10.1016/j.crma.2006.12.016
[7] DOI: 10.1007/BF01211563 · Zbl 0760.93067 · doi:10.1007/BF01211563
[8] Coron J.-M., C. R. Acad. Sci. Paris Sér. I Math. 317 pp 271–
[9] DOI: 10.1051/cocv:1996102 · Zbl 0872.93040 · doi:10.1051/cocv:1996102
[10] Coron J.-M., J. Math. Pures Appl. (9) 75 pp 155–
[11] Coron J.-M., Mathematical Surveys and Monographs 136, in: Control and Nonlinearity (2007)
[12] Coron J.-M., J. Eur. Math. Soc. (JEMS) 6 pp 367–
[13] Coron J.-M., Russian J. Math. Phys. 4 pp 429–
[14] Dauxois T., Physics of Solitons (2006) · Zbl 1192.35001
[15] DOI: 10.1016/S0764-4442(97)89091-X · Zbl 0897.76014 · doi:10.1016/S0764-4442(97)89091-X
[16] DOI: 10.1051/cocv:2000100 · Zbl 0940.93012 · doi:10.1051/cocv:2000100
[17] Glass O., Asymptot. Anal. 60 pp 61–
[18] DOI: 10.1080/14786449508620739 · doi:10.1080/14786449508620739
[19] Lions J. L., Travaux et recherches mathématiques 1, in: Problèmes aux limites non homogènes et applications (1968)
[20] DOI: 10.1051/cocv:2005015 · Zbl 1148.35348 · doi:10.1051/cocv:2005015
[21] Menzala G. P., Quart. Appl. Math. 60 pp 111–
[22] DOI: 10.1051/cocv:1997102 · Zbl 0873.93008 · doi:10.1051/cocv:1997102
[23] DOI: 10.1137/S0363012999353229 · Zbl 0966.93055 · doi:10.1137/S0363012999353229
[24] DOI: 10.1051/cocv:2004012 · Zbl 1094.93014 · doi:10.1051/cocv:2004012
[25] DOI: 10.1137/050631409 · Zbl 1116.35108 · doi:10.1137/050631409
[26] DOI: 10.1090/S0002-9947-96-01672-8 · Zbl 0862.93035 · doi:10.1090/S0002-9947-96-01672-8
[27] Whitham G. B., Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs and Tracts, in: Linear and Nonlinear Waves (1974)
[28] DOI: 10.1137/S0363012997327501 · Zbl 0930.35160 · doi:10.1137/S0363012997327501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.