zbMATH — the first resource for mathematics

Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. (English) Zbl 0873.93008
The author proves a controllability result for the nonlinear Korteweg-de Vries equation on bounded domains for sufficiently small initial and final states. He uses the Banach contraction fixed point theorem and the corresponding controllability result for the linear equation. For the linear case, the author uses the Hilbert Uniqueness Method and the multiplier method.
Reviewer: O.Cârjá (Iaşi)

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
Full Text: DOI Link EuDML
[1] C. Bardos, G. Lebeau and J. Rauch: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 1992, 1024-1065. Zbl0786.93009 MR1178650 · Zbl 0786.93009 · doi:10.1137/0330055
[2] J. Bona and R. Winther: The Korteweg-de Vries equation, posed in a quarter-plane, SIAM J. Math Anal., 14, 1983, 1056-1106. Zbl0529.35069 MR718811 · Zbl 0529.35069 · doi:10.1137/0514085
[3] J.M. Coron: Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris, t. 317, Série I, 1993, 271-276. Zbl0781.76013 MR1233425 · Zbl 0781.76013
[4] A.V. Fursikov and O. Y. Imanuvilov: On controllability of certain systems simulating a fluid flow, in Flow Control, IMA, Math. Appl., vol. 68, Gunzberger ed., Springer-Verlag, New York, 1995, 148-184. Zbl0922.93006 MR1348646 · Zbl 0922.93006
[5] L.F. Ho: Observabilité frontière de l’équation des ondes, C. R. Acad. Sci. Paris, Série 1 Math., 302, 1986, 443-446. Zbl0598.35060 MR838598 · Zbl 0598.35060
[6] A.E. Ingham: Some trigonometrical inequalities with application to the theory of series, Math. A., 41, 1936, 367-379. Zbl0014.21503 MR1545625 · Zbl 0014.21503 · doi:10.1007/BF01180426 · eudml:168670
[7] V. Komornik: Exact controllability and stabilization, the multiplier method, R.A.M. 36, John Wiley-Masson, 1994. Zbl0937.93003 MR1359765 · Zbl 0937.93003
[8] V. Komornik, D.L. Russel and B.-Y. Zhang: Control and stabilization of the Korteweg-de Vries equation on a periodic domain, submitted to J. Differential Equations. · Zbl 1213.93015 · doi:10.1080/03605300903585336
[9] D.J. Korteweg and G. de Vries: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 5, 39, 1895, 422-423. JFM26.0881.02 · JFM 26.0881.02
[10] G. Lebeau: Contrôle de l’équation de Schrödinger, J. Math. Pures Appl., 71, 1992, 267-291. Zbl0838.35013 MR1172452 · Zbl 0838.35013
[11] J.L. Lions: Contrôlabilité exacte de systèmes distribués, C.R. Acad. Sci. Paris, 302, 1986, 471-475. Zbl0589.49022 MR838402 · Zbl 0589.49022
[12] J.L. Lions: Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité exacte, Collection de recherche en mathématiques appliquées, 8, Masson, Paris, 1988. Zbl0653.93002 MR953547 · Zbl 0653.93002
[13] J.L. Lions: Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30, 1988, 1-68. Zbl0644.49028 MR931277 · Zbl 0644.49028 · doi:10.1137/1030001
[14] E. Machtyngier: Exact controllability for the Schrödinger equation, SIAM J. Control Optim., 32, 1994, 24-34. Zbl0795.93018 MR1255957 · Zbl 0795.93018 · doi:10.1137/S0363012991223145
[15] A. Pazy: Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. Zbl0516.47023 MR710486 · Zbl 0516.47023
[16] D.L. Russel and B-Y. Zhang: Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31, 1993, 659-673. Zbl0771.93073 MR1214759 · Zbl 0771.93073 · doi:10.1137/0331030
[17] J.C. Saut and R. Temam: Remarks on the Korteweg-De Vries equation, Israel. Math., 24, 1976, 78-87. Zbl0334.35062 MR454425 · Zbl 0334.35062 · doi:10.1007/BF02761431
[18] J. Simon: Compact sets in the space Lp(0,T, B), Annali di Matematica pura ed applicata (IV), vol. CXLVI, 1987, 65-96. Zbl0629.46031 MR916688 · Zbl 0629.46031 · doi:10.1007/BF01762360
[19] K. Yosida: Functional Analysis, Springer-Verlag, Berlin Heidelberg New York, 1978. Zbl0365.46001 MR500055 · Zbl 0365.46001
[20] B.-Y. Zhang: Some results for nonlinear dispersive wave equations with applications to control, Ph. D. thesis, University of Wisconsin, Madison, June 1990.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.