Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T05:23:28.518Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact boundary observability for quasilinear hyperbolicsystems

Published online by Cambridge University Press:  30 January 2008

Tatsien Li Daqian Li
Tatsien Li Daqian Li*
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, China; [email protected]
Get access

Abstract

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.

Keywords

Exact boundary observabilityexact boundarycontrollabilitysemi-global C 1 solutionmixedinitial-boundary value problemquasilinear hyperbolic system.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 759 - 766
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alabau, F. and Komornik, V., Observabilité, contrôlabilité et stabilisation frontière du système d'élasticité linéaire. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 519–524. CrossRef
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. CrossRef
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.
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.
Li, T. and Rao, B., Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin. Ann. Math. 23B (2002) 209–218. CrossRef
Li, T. and Rao, B., Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control Optim. 41 (2003) 1748–1755. CrossRef
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome I: Contrôlabilité Exacte, RMA 8. Masson (1988).
Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. CrossRef
I. Trooshin and M. Yamamoto, Identification problem for a one-dimensional vibrating system. Math. Meth. Appl. Sci. 28 (2005) 2037–2059.
Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chin. Ann. Math. 27B (2006) 643–656.
P. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37 (1999) 1568–1599.
Zuazua, E., Boundary observability for the space-discretization of the 1-D wave equation. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 713–718.
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.