Parameter estimation and output feedback stabilization for the linear Korteweg-de Vries equation with disturbed boundary measurement
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 76.

This paper is concerned with the parameter estimation and boundary feedback stabilization for the linear Korteweg-de Vries equation posed on a finite interval with the boundary observation at the right end and the non-collocated control at the left end. The boundary observation suffers from some unknown disturbance. An adaptive observer is designed and the adaptive laws of the parameters are obtained by the Lyapunov method. The resulted closed-loop system is proved to be well-posed and asymptotically stable in case that the length of the interval is not critical. Moreover, it is shown that the estimated parameter converges to the unknown parameter. As a by-product, a hidden regularity result is proved.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018057
Classification : 35B35, 35Q53, 93C20
Mots-clés : Korteweg-de Vries equation, output feedback stabilization, adaptive observer, hidden regularity
Jia, Chaohua 1 ; Guo, Wei 1 ; Luo, Diao 1

1 
@article{COCV_2019__25__A76_0,
     author = {Jia, Chaohua and Guo, Wei and Luo, Diao},
     title = {Parameter estimation and output feedback stabilization for the linear {Korteweg-de} {Vries} equation with disturbed boundary measurement},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {25},
     year = {2019},
     doi = {10.1051/cocv/2018057},
     mrnumber = {4039138},
     zbl = {1437.35075},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2018057/}
}
TY  - JOUR
AU  - Jia, Chaohua
AU  - Guo, Wei
AU  - Luo, Diao
TI  - Parameter estimation and output feedback stabilization for the linear Korteweg-de Vries equation with disturbed boundary measurement
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2019
VL  - 25
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2018057/
DO  - 10.1051/cocv/2018057
LA  - en
ID  - COCV_2019__25__A76_0
ER  - 
%0 Journal Article
%A Jia, Chaohua
%A Guo, Wei
%A Luo, Diao
%T Parameter estimation and output feedback stabilization for the linear Korteweg-de Vries equation with disturbed boundary measurement
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2019
%V 25
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2018057/
%R 10.1051/cocv/2018057
%G en
%F COCV_2019__25__A76_0
Jia, Chaohua; Guo, Wei; Luo, Diao. Parameter estimation and output feedback stabilization for the linear Korteweg-de Vries equation with disturbed boundary measurement. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 76. doi : 10.1051/cocv/2018057. http://www.numdam.org/articles/10.1051/cocv/2018057/

[1] J.L. Bona, S.-M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Comm. Partial Differ. Equ. 28 (2003) 1391–1436. | DOI | MR | Zbl

[2] J.L. Bona, S.-M. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane. Ann. Inst. Henri Poincaré 25 (2008) 1145–1185. | DOI | Numdam | MR | Zbl

[3] E. Cerpa and J.-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from left Dirichlet boundary condition. IEEE Trans. Auto. Control 58 (2013) 1688–1695. | DOI | MR | Zbl

[4] E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 655–668. | MR | Zbl

[5] J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right. J. Math. Pures Appl. 102 (2014) 1080–1120. | DOI | MR | Zbl

[6] M.G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets. J. Funct. Anal. 3 (1969) 376–418. | DOI | MR | Zbl

[7] C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97–106. | DOI | MR | Zbl

[8] W. Guo and B.-Z. Guo, Parameter estimation and stabilization for a one-dimensional wave equation with boundary output constant disturbance and non-collocated control. Int. J. Control 84 (2011) 381–395. | DOI | MR | Zbl

[9] B.-Z. Guo, H.-C. Zhou, A.S. Al-Fhaid, A.M.M. Younas and A. Asiri, Parameter estimation and stabilization for one-dimensional Schrödinger equation with boundary output constant disturbance and non-collocated control. J. Franklin Inst. 352 (2015) 2047–2064. | DOI | MR | Zbl

[10] A. Hasan, Output-Feedback Stabilization of the Korteweg-de Vries Equation. 24th Mediterranean Conference on Control and Automation (MED), Athens, Greece (2016) 871–876.

[11] C.-H. Jia, Boundary feedback stabilization of the Korteweg-de Vries-Burgers equation posed on a finite interval. J. Math. Anal. Appl. 444 (2016) 624–47. | DOI | MR | Zbl

[12] C.-H. Jia and B.-Y. Zhang, Boundary stabilization of the Korteweg-de Vries equation and the Korteweg-de Vries-Burgers equation. Acta Appl. Math. 118 (2012) 25–47. | DOI | MR | Zbl

[13] E. Kramer, I. Rivas and B.-Y. Zhang, Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain. ESAIM: COCV 19 (2013) 358–384. | Numdam | MR | Zbl

[14] M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs – part I: Lyapunov design. IEEE Trans. Autom. Control 53 (2008) 1575–1591. | DOI | MR | Zbl

[15] Z.H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag, London (1998). | MR | Zbl

[16] S. Marx and E. Cerpa, Output feedback stabilization of the Korteweg-de Vries equation. Automatica 87 (2018) 210–217. | DOI | MR | Zbl

[17] S. Marx and E. Cerpa, Output feedback control of the linear Korteweg-de Vries equation. 53rd IEEE Conference on Decision and Control, Los Angeles, USA (2014) 2083–2087. | DOI

[18] 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

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

[20] S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries systems with anti-diffusion. 2013 American Control Conference, Washington DC, USA (2013) 3302–3307. | DOI

[21] S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries systems with anti-diffusion by boundary feedback with non-collocated observation. 2015 American Control Conference, Palmer House Hilton, USA (2015) 1959–1964. | DOI

[22] G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17–43. | DOI | MR | Zbl

[23] G. Weiss, Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527–545. | DOI | MR | Zbl

[24] B.-Y. Zhang, Boundary stabilization of the Korteweg-de Vries equations. Int. Series Numer. Math. 118 (1994) 371–389. | MR | Zbl

Cité par Sources :