This paper deals with the problem of regional observability of hyperbolic systems in the case where the subregion of interest is a boundary part of the system evolution domain. We give a definition and establish characterizations in connection with the sensor structure. Then we show that it is possible to reconstruct the system state on a subregion of the boundary. The developed approach, based on the Hilbert uniqueness method (Lions, 1988), leads to a reconstruction algorithm. The obtained results are illustrated with numerical examples and simulations.
@article{bwmeta1.element.bwnjournal-article-amcv20i2p227bwm,
author = {El Hassan Zerrik and Hamid Bourray and Samir Ben Hadid},
title = {Sensors and boundary state reconstruction of hyperbolic systems},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {20},
year = {2010},
pages = {227-238},
zbl = {1196.93012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv20i2p227bwm}
}
El Hassan Zerrik; Hamid Bourray; Samir Ben Hadid. Sensors and boundary state reconstruction of hyperbolic systems. International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) pp. 227-238. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv20i2p227bwm/
[000] Amouroux, A., El Jai, M. and Zerrik, E. (1994). Regional observability of distributed systems, International Journal of Systems Science 25(2): 301-313. | Zbl 0812.93015
[001] Avdonin, S.A. and Ivanov, S.A. (1995). Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, NY/London/Melbourne. | Zbl 0866.93001
[002] Avdonin, S.A. and Ivanov, S.A. (1995). Boundary controllability problems for the wave equation in a parallelepiped, Applied Mathematic Letters 8(2): 97-102. | Zbl 0822.93011
[003] Avdonin, S.A., Ivanov, S.A. and Joó, I. (1995). Exponential series in the problem of initial and pointwise control of a rectangular vibrating membrane, Studia Scientiarium Mathematicarum Hungarica 30(3-4): 243-259. | Zbl 0863.93041
[004] Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, Vol. 21, Springer-Verlag, New York, NY. | Zbl 0839.93001
[005] Curtain, R.F. and Pritchard, A.J. (1978). Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, NY. | Zbl 0389.93001
[006] Dolecki, S. and Russel, D. (1977). A general theory of observation and control, SIAM Journal of Control 15(2): 185-220. | Zbl 0353.93012
[007] El Jai, A. and Pritchard, A.J. (1988). Sensors and Actuators in Distributed Systems Analysis, J. Wiley, New York, NY. | Zbl 0637.93001
[008] Kobayashi, T. (1980). Discrete-time observability for distributed parameter systems, International Journal of Control 31(1): 181-193. | Zbl 0462.93011
[009] Lions, J.L. and Magenes, E. (1968). Problèmes aux limites non homogènes et applications, Vols. 1 et 2, Dunod, Paris. | Zbl 0165.10801
[010] Lions, J.L. (1968). Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris. | Zbl 0179.41801
[011] Lions, J.L. (1988). Contrôlabilité Exacte. Perturbations et Stabilisation des Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson, Paris. | Zbl 0653.93002
[012] Li, D., Gilliam, Z. and Martin, C. (1988). Discrete observavility of the heat equation on bounded domain, International Journal of Control 48(2): 755-780. | Zbl 0654.93008
[013] Micheletti, A. M. (1976). Perturbazione dello specttro di Un operatore ellitico di tipo variazionale, in relazione ad Una variazione del compo, Ricerche di matematica, V. XXV, Fasc. II. | Zbl 0355.35066
[014] Zerrik, E., Bourray, H. and Boutoulout, A. (2002). Regional boundary observability, numerical approach, International Journal of Applied Mathematics and Computer Science 12(2): 143-151. | Zbl 1140.93328
[015] Zerrik, E., Ben Hadid, S. and Bourray, H. (2007). Sensors and regional observability of hyperbolic systems, Sensors and Actuator Journal 138(2): 313-328.