Exact boundary observability for quasilinear hyperbolic systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 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 : https://doi.org/10.1051/cocv:2008007
Classification : 35B37,  93C20,  35L50,  93B07,  35R30
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},
     zbl = {1155.93015},
     mrnumber = {2451794},
     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
DA  - 2008///
SP  - 759
EP  - 766
VL  - 14
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008007/
UR  - https://zbmath.org/?q=an%3A1155.93015
UR  - https://www.ams.org/mathscinet-getitem?mr=2451794
UR  - https://doi.org/10.1051/cocv:2008007
DO  - 10.1051/cocv:2008007
LA  - en
ID  - COCV_2008__14_4_759_0
ER  - 
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] 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 1443987 | Zbl 0878.73012

[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 1178650 | Zbl 0786.93009

[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 1758667 | Zbl 0931.35022

[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 1845753 | Zbl 1005.35058

[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 1924137 | Zbl 1184.35196

[6] T. Li and B. Rao, Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control Optim. 41 (2003) 1748-1755. | MR 1972532 | Zbl 1032.35124

[7] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome I: Contrôlabilité Exacte, RMA 8. Masson (1988). | Zbl 0653.93002

[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 508380 | Zbl 0397.93001

[9] I. Trooshin and M. Yamamoto, Identification problem for a one-dimensional vibrating system. Math. Meth. Appl. Sci. 28 (2005) 2037-2059. | MR 2176906 | Zbl 1083.35136

[10] Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chin. Ann. Math. 27B (2006) 643-656. | MR 2273803

[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 1710233 | Zbl 0951.35069

[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 1641762 | Zbl 0910.65051

[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 1697041 | Zbl 0939.93016

Cité par Sources :