By means of a direct and constructive method based on the theory of semi-global solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.
Mots-clés : exact boundary observability, exact boundary controllability, semi-global $C^1$ solution, mixed initial-boundary value problem, quasilinear hyperbolic system
@article{COCV_2008__14_4_759_0,
author = {Tatsien Li Daqian Li},
title = {Exact boundary observability for quasilinear hyperbolic systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {759--766},
publisher = {EDP-Sciences},
volume = {14},
number = {4},
year = {2008},
doi = {10.1051/cocv:2008007},
mrnumber = {2451794},
zbl = {1155.93015},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008007/}
}
TY - JOUR AU - Tatsien Li Daqian Li TI - Exact boundary observability for quasilinear hyperbolic systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 759 EP - 766 VL - 14 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008007/ DO - 10.1051/cocv:2008007 LA - en ID - COCV_2008__14_4_759_0 ER -
%0 Journal Article %A Tatsien Li Daqian Li %T Exact boundary observability for quasilinear hyperbolic systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 759-766 %V 14 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008007/ %R 10.1051/cocv:2008007 %G en %F COCV_2008__14_4_759_0
Tatsien Li Daqian Li. Exact boundary observability for quasilinear hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 759-766. doi : 10.1051/cocv:2008007. http://www.numdam.org/articles/10.1051/cocv:2008007/
[1] and , Observabilité, contrôlabilité et stabilisation frontière du système d'élasticité linéaire. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 519-524. | MR | Zbl
[2] , and , Sharp efficient conditions for the observation, control and stabilization of wave from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR | Zbl
[3] , and , Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235 (1999) 13-57. | MR | Zbl
[4] and , Semi-global solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. 22B (2001) 325-336. | MR | Zbl
[5] and , Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin. Ann. Math. 23B (2002) 209-218. | MR | Zbl
[6] and , Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control Optim. 41 (2003) 1748-1755. | MR | Zbl
[7] , Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome I: Contrôlabilité Exacte, RMA 8. Masson (1988). | Zbl
[8] , Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639-739. | MR | Zbl
[9] and , Identification problem for a one-dimensional vibrating system. Math. Meth. Appl. Sci. 28 (2005) 2037-2059. | MR | Zbl
[10] , Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chin. Ann. Math. 27B (2006) 643-656. | MR
[11] , On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37 (1999) 1568-1599. | MR | Zbl
[12] , Boundary observability for the space-discretization of the 1- wave equation. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 713-718. | MR | Zbl
[13] , Boundary observability for the finite-difference space semi-discretizations of the 2- wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. | MR | Zbl
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