Internal controllability of the korteweg–de vries equation on a bounded domain
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 1076-1107.

This paper is concerned with the control properties of the Korteweg–de Vries (KdV) equation posed on a bounded interval (0,L) with a distributed control. When the control region is an arbitrary open subdomain (l 1 ,l 2 ), we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, which roughly tells us that any target function arbitrarily chosen on (0,l 1 ) and null on (l 2 ,L) is reachable. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L 2 -space is also established.

Received:
DOI: 10.1051/cocv/2014059
Classification: 35Q53, 37K10, 93B05, 93D15
Keywords: KdV equation, Carleman estimate, null controllability, exact controllability
Capistrano–Filho, Roberto A. 1, 2; Pazoto, Ademir F. 1; Rosier, Lionel 3

1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária, Ilha do Fundão, 21941-909 Rio de Janeiro (RJ), Brazil.
2 Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, BP 70239, 54506 Vandœuvre-les-Nancy cedex, France.
3 Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 boulevard Saint-Michel, 75272 Paris cedex 06, France.
@article{COCV_2015__21_4_1076_0,
     author = {Capistrano{\textendash}Filho, Roberto A. and Pazoto, Ademir F. and Rosier, Lionel},
     title = {Internal controllability of the korteweg{\textendash}de vries equation on a bounded domain},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1076--1107},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {4},
     year = {2015},
     doi = {10.1051/cocv/2014059},
     mrnumber = {3395756},
     zbl = {1331.35302},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2014059/}
}
TY  - JOUR
AU  - Capistrano–Filho, Roberto A.
AU  - Pazoto, Ademir F.
AU  - Rosier, Lionel
TI  - Internal controllability of the korteweg–de vries equation on a bounded domain
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 1076
EP  - 1107
VL  - 21
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2014059/
DO  - 10.1051/cocv/2014059
LA  - en
ID  - COCV_2015__21_4_1076_0
ER  - 
%0 Journal Article
%A Capistrano–Filho, Roberto A.
%A Pazoto, Ademir F.
%A Rosier, Lionel
%T Internal controllability of the korteweg–de vries equation on a bounded domain
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 1076-1107
%V 21
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2014059/
%R 10.1051/cocv/2014059
%G en
%F COCV_2015__21_4_1076_0
Capistrano–Filho, Roberto A.; Pazoto, Ademir F.; Rosier, Lionel. Internal controllability of the korteweg–de vries equation on a bounded domain. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 1076-1107. doi : 10.1051/cocv/2014059. http://www.numdam.org/articles/10.1051/cocv/2014059/

R.A. Adams, Sobolev Spaces, 1st edition. Academic Press, New York, San Francisco London, 1st edition (1975). | MR | Zbl

J. Bergh and J. Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematishen Wissenschaften, No. 223. Springer-Verlag, Berlin New York (1976). | MR | Zbl

J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l’Acad. Sci. Inst. Nat. France 23 (1877) 1–680. | JFM

E. Cerpa, Exact controllability of a nonlinear Korteweg–de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46 (2007) 877–899. | DOI | MR | Zbl

E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain. Ann. Institut Henri Poincaré 26 (2009) 457–475. | DOI | Numdam | MR | Zbl

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length. J. Eur. Math. Soc. 6 (2004) 367–398. | DOI | MR | Zbl

S. Dolecki and D.L. Russell, A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185–220. | DOI | MR | Zbl

O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptot. Anal. 60 (2008) 61–100. | MR | Zbl

O. Glass and S. Guerrero, Controllability of the Korteweg–de Vries equation from the right Dirichlet boundary condition. Systems Control Lett. 59 (2010) 390–395. | DOI | MR | Zbl

O. Goubet and J. Shen, On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval. Adv. Differ. Equ. 12 (2007) 221–239. | MR | Zbl

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations. Vol. 26 of Math. Appl. Springer-Verlag (1997). | MR | Zbl

C.E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996) 573–603. | DOI | MR | Zbl

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. 39 (1895) 422–443. | DOI | JFM | MR

C. Laurent, L. Rosier and B.Y. Zhang, Control and stabilization of the Korteweg–de Vries equation on a periodic domain. Commun. Partial Differ. Equ. 35 (2010) 707–744. | DOI | MR | Zbl

J.-L Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Reviews 30 (1988) 1–68. | DOI | MR | Zbl

J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1 of Travaux et Recherches Mathématiques. Dunod, Paris (1968) 17. | Zbl

A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights. Inverse Prob. 24 (2008) 015017. | DOI | MR | Zbl

R.M. Miura, The Korteweg–de Vries equation: A survey of results. SIAM Reviews 18 (1976) 412–459. | DOI | MR | Zbl

G. Perla-Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg–de Vries equation with localized damping. Quart. Appl. Math. 60 (2002) 111–129. | DOI | MR | Zbl

L. Rosier, Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. | Numdam | MR | Zbl

L. Rosier, Exact boundary controllability for the linear Korteweg–de Vries equation on the half-line. SIAM J. Control Optim. 39 (2000) 331–351. | DOI | MR | Zbl

L. Rosier, Control of the surface of a fluid by a wavemaker. ESAIM: COCV 10 (2004) 346–380. | Numdam | MR | Zbl

L. Rosier and B.-Y. Zhang, Null controllability of the complex Ginzburg–Landau equation. Ann. Institut Henri Poincaré Anal. Non Linéaire 26 (2009) 649–673. | DOI | Numdam | MR | Zbl

L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg–de Vries equation: Recent progresses. J. Syst. Sci. Complexity 22 (2009) 647–682. | DOI | MR | Zbl

D.L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg–de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 3643–3672. | DOI | MR | Zbl

J.-C. Saut and R. Temam, Remarks on the Korteweg–de Vries equation. Israel J. Math. 24 (1976) 78–87. | DOI | MR | Zbl

E. Zeidler, Nonlinear functional analysis and its applications I. Springer-Verlag, New York (1986). | MR | Zbl

B.-Y. Zhang, Exact boundary controllability of the Korteweg–de Vries equation. SIAM J. Cont. Optim. 37 (1999) 543–565. | DOI | MR | Zbl

Cited by Sources: