Exact boundary observability for quasilinear hyperbolic systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 4, pp. 759-766.

By means of a direct and constructive method based on the theory of semi-global C 1 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.

DOI: 10.1051/cocv:2008007
Classification: 35B37, 93C20, 35L50, 93B07, 35R30
Keywords: 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://archive.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://archive.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://archive.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, Volume 14 (2008) no. 4, pp. 759-766. doi : 10.1051/cocv:2008007. http://archive.numdam.org/articles/10.1051/cocv:2008007/

[1] F. Alabau and V. Komornik, 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] C. Bardos, G. Lebeau and R. Rauch, 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] I. Lasiecka, R. Triggiani and P. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235 (1999) 13-57. | MR | Zbl

[4] T. Li and Y. Jin, Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. 22B (2001) 325-336. | MR | Zbl

[5] T. Li and B. Rao, Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin. Ann. Math. 23B (2002) 209-218. | MR | Zbl

[6] T. Li and B. Rao, Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control Optim. 41 (2003) 1748-1755. | MR | Zbl

[7] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome I: Contrôlabilité Exacte, RMA 8. Masson (1988). | Zbl

[8] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639-739. | MR | Zbl

[9] I. Trooshin and M. Yamamoto, Identification problem for a one-dimensional vibrating system. Math. Meth. Appl. Sci. 28 (2005) 2037-2059. | MR | Zbl

[10] Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chin. Ann. Math. 27B (2006) 643-656. | MR

[11] P. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37 (1999) 1568-1599. | MR | Zbl

[12] E. Zuazua, Boundary observability for the space-discretization of the 1-D wave equation. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 713-718. | MR | Zbl

[13] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. | MR | Zbl

Cited by Sources: