Abstract
In this paper, we consider the problem of exact boundary controllability of a linear Korteweg-de Vries (KdV) equation in a bounded domain when the condition for the control is the difference between the derivative of the solution in the left and right endpoint. We prove the existence of a countable set of critical lengths out of which we have the exact controllability. In the second part of this paper, we study the behavior of the optimal control and how the cost of controllability evolves as the dispersive term brought to zero.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brezis H., Analyse Fonctionnelle: Théorie et Application, 5ième édition, Masson.
Cerpa E. Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., Vol. 46, No. 3, (2007), pp. 877–899
Cerpa E., Crépeau E. Boundary controllabilty for the nonlinear Korteweg-de Vries equation on any critical domain, Annales Institut Henri Poincaré, Analyse non lineaire, 26 (2) (2009) pp 457–475.
Coron J.M., Control and nonlinearity, Mathematical Surveys and Monographs, 136, 2007, 427
Coron J.-M., Crépeau E., Exact boundary controllability of a nonlinear KdV equation with a critical lengths J.Eur.Math.Soc. vol 6, (2004), pp. 367–398.
Glass O., Guerrero S., Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptotic Analysis 60 (2008) pp 61–100.
Ingham A. E., Some trigonometrical inequality with applications to the theory of series, Math.,Zeitschrift, 41,(1936), 367–379.
Jaffard S., Micu S., Estimates of the constants in generalized Ingham’s inequality and applications to the control of the wave equation, Asymptotic Analysis Vol. 28, No.3–4, (2001), pp. 181–214
Lions J.L., Contrôlabilité exacte, Perturbations et Stabilization de systèmes distribués. Tome 1, Recherche en mathématiques appliquées [Research in Applied Mathematics], vol.8, (Masson, Paris), (1988).
Rosier L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var.2 (1997), 33–55.
Rosier L., Control of the surface of a fluid by a wavemaker, ESAIM: Control, Optimisation and Calculus of Variations, vol 10, (2004), pp 346–380.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Dbebria, H., Salem, A. (2015). Exact Controllability For Korteweg-De Vries Equation and its Cost in the Zero-Dispersion Limit. In: Jeribi, A., Hammami, M., Masmoudi, A. (eds) Applied Mathematics in Tunisia. Springer Proceedings in Mathematics & Statistics, vol 131. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18041-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-18041-0_19
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-18040-3
Online ISBN: 978-3-319-18041-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)