June  2015, 4(2): 177-192. doi: 10.3934/eect.2015.4.177

Flux reconstruction for hyperbolic systems: Sensors and simulations

1. 

TSI Team, MACS Laboratory, Moulay Ismail University, Faculty of Sciences, PB 11201, Zitoune-Meknes, Morocco, Morocco

Received  April 2014 Revised  October 2014 Published  May 2015

This paper aims to establish necessary conditions for sensors structure (number and location) in order to obtain regional boundary gradient observability for hyperbolic system. The obtained results are applied to a two-dimensional diffusion process considering various types of sensors. Also, we will explore an approach that can reconstruct the gradient on a part $\Gamma$ of the boundary $\partial\Omega$ of the evolution domain $\Omega$. The simulations illustrate the established results and lead to some conjectures.
Citation: Adil Khazari, Ali Boutoulout. Flux reconstruction for hyperbolic systems: Sensors and simulations. Evolution Equations and Control Theory, 2015, 4 (2) : 177-192. doi: 10.3934/eect.2015.4.177
References:
[1]

A. Boutoulout, H. Bourray and A. Khazari, Gradient observability for hyperbolic system, International Review of Automatic Control, 6 (2013), 274-263.

[2]

A. Boutoulout, H. Bourray and A. Khazari, Flux observability for hyperbolic systems, Appl. Math. Inf. Sci. Lett., 2 (2014), 13-24.

[3]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer Lecture Notes in Control and Informations, Science, Springer, New York, 1978.

[4]

R. F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[5]

A. El Jai and A. J. Pritchard, Sensors and actuators in distributed systems analysis, Internat. J. Control, 46 (1987), 1139-1153. doi: 10.1080/00207178708933956.

[6]

A. El Jai, M. C. Simon and E. Zerrik, Regional observability and sensor structures, Sensors and Actuators Journal, 39 (1993), 95-102.

[7]

A. El Jai, M. Amouroux and E. Zerrik, Regional observability of distributed systems, Int. J. Syst. Sci., 25 (1994), 301-313. doi: 10.1080/00207729408928961.

[8]

A. M. Micheletti, Perturbazione Dello Spettro di un Opertore Ellitico di Tipo Variazionale, in Relazione ad una Variazione del Compo, Ricerche di matematica, in Italian, XXV, Fasc II, 1976.

[9]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson, Paris, 1988.

[10]

J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications. Vol. 1, Travaux et Recherches Mathmatiques, No. 17, Dunod, Paris, 1968.

[11]

E. Zerrik, A. Boutoulout and A. El Jai, Actuators and regional boundary controllability of parabolic systems, Int. J. Syst. Sci., 31 (2000), 73-82. doi: 10.1080/002077200291479.

[12]

E. Zerrik and H. Bourray, Gradient observability for diffustion system, Int. J. Appl. Math. Comput. Sci., 13 (2003), 139-150.

[13]

E. Zerrik and H. Bourray, Flux reconstruction: Sensors and simulations, Sensors and Actuators A, 109 (2003), 34-46. doi: 10.1016/S0924-4247(03)00358-3.

[14]

E. Zerrik, H. Bourray and S. Benhadid, Sensors and Regional observability of the wave equation, Sensors and Actuators Journal, 138 (2007), 313-328. doi: 10.1016/j.sna.2007.05.017.

[15]

E. Zerrik, H. Bourray and S. Benhadid, Sensors and boundary state reconstruction of hyperbolic systems, Int. J. Appl. Math. Comput. Sci., 20 (2010), 227-238. doi: 10.2478/v10006-010-0016-4.

show all references

References:
[1]

A. Boutoulout, H. Bourray and A. Khazari, Gradient observability for hyperbolic system, International Review of Automatic Control, 6 (2013), 274-263.

[2]

A. Boutoulout, H. Bourray and A. Khazari, Flux observability for hyperbolic systems, Appl. Math. Inf. Sci. Lett., 2 (2014), 13-24.

[3]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer Lecture Notes in Control and Informations, Science, Springer, New York, 1978.

[4]

R. F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[5]

A. El Jai and A. J. Pritchard, Sensors and actuators in distributed systems analysis, Internat. J. Control, 46 (1987), 1139-1153. doi: 10.1080/00207178708933956.

[6]

A. El Jai, M. C. Simon and E. Zerrik, Regional observability and sensor structures, Sensors and Actuators Journal, 39 (1993), 95-102.

[7]

A. El Jai, M. Amouroux and E. Zerrik, Regional observability of distributed systems, Int. J. Syst. Sci., 25 (1994), 301-313. doi: 10.1080/00207729408928961.

[8]

A. M. Micheletti, Perturbazione Dello Spettro di un Opertore Ellitico di Tipo Variazionale, in Relazione ad una Variazione del Compo, Ricerche di matematica, in Italian, XXV, Fasc II, 1976.

[9]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson, Paris, 1988.

[10]

J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications. Vol. 1, Travaux et Recherches Mathmatiques, No. 17, Dunod, Paris, 1968.

[11]

E. Zerrik, A. Boutoulout and A. El Jai, Actuators and regional boundary controllability of parabolic systems, Int. J. Syst. Sci., 31 (2000), 73-82. doi: 10.1080/002077200291479.

[12]

E. Zerrik and H. Bourray, Gradient observability for diffustion system, Int. J. Appl. Math. Comput. Sci., 13 (2003), 139-150.

[13]

E. Zerrik and H. Bourray, Flux reconstruction: Sensors and simulations, Sensors and Actuators A, 109 (2003), 34-46. doi: 10.1016/S0924-4247(03)00358-3.

[14]

E. Zerrik, H. Bourray and S. Benhadid, Sensors and Regional observability of the wave equation, Sensors and Actuators Journal, 138 (2007), 313-328. doi: 10.1016/j.sna.2007.05.017.

[15]

E. Zerrik, H. Bourray and S. Benhadid, Sensors and boundary state reconstruction of hyperbolic systems, Int. J. Appl. Math. Comput. Sci., 20 (2010), 227-238. doi: 10.2478/v10006-010-0016-4.

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