Abstract
In this paper, we deal with the null controllability of a linear Korteweg-de Vries equation with special boundary conditions. We first study the Cauchy problem and prove the existence and uniqueness of solutions in suitable spaces, and then prove the null controllability of this system by establishing an observability inequality which is proved with the help of a Carleman estimate.
Similar content being viewed by others
References
Bona J, Sun SM, Zhang B-Y (2003) A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Comm Partial Differ Eq 28(7–8):1391–1436
Cerpa E (2007) Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J Control Optim 46:877–899
Cerpa E, Crépeau E (2009) Boundary controllability for the non linear Korteweg-de Vries equation on any critical domain. Ann Inst H Poincaré Anal Non Linéaire 26(2):457–475
Colin T, Ghidaglia J-M (1997) Un problème aux limites pour l’équation de Korteweg-de Vries sur un intervalle borné. (French) [A boundary value problem for the Korteweg-de Vries equation on a bounded interval] Journées “Equations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1997), Exp. No. III, p 10, Ecole Polytech, Palaiseau, 35Q53
Colin T, Ghidaglia J-M (2001) Jean-Michel An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval. Adv Differ Equ 6(12):1463–1492
Coron J-M (2007) Control and nonlinearity’ Mathematical Surveys and Monographs, vol 136. American Mathematical Society, Providence
Coron J-M, Crépeau E (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths. J Eur Math Soc 6:367–398
Glass O, Guerrero S (2008) Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptot Anal 60(1–2):61–100
Holmer J (2006) The initial-boundary value problem for the Korteweg-de Vries equation. Comm Partial Differ Eq 31(7–9):1151–1190
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations, applied Mathematical Sciences, vol 44. Springer, New York
Rosier L (1997) Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim Calc Var 2:33–55
Rosier L (2000) Lionel Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line. SIAM J Control Optim 39(2):331–351
Rosier L (2004) Control of the surface of a fluid by a wavemaker. ESAIM Control Optim Calc Var 10(3):346–380
Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, Texts in applied mathematics, vol 21. Springer, New York
Cerpa E, Rivas Y, Zhang B-Y (2013) Boundary controllability of the Korteweg-de Vries equation on a bounded domain. SIAM J Control Optim 51(4):2976–3010
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guilleron, JP. Null controllability of a linear KdV equation on an interval with special boundary conditions. Math. Control Signals Syst. 26, 375–401 (2014). https://doi.org/10.1007/s00498-013-0122-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-013-0122-6