American Institute of Mathematical Sciences

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

Flux reconstruction for hyperbolic systems: Sensors and simulations

Adil Khazari 1, and  Ali Boutoulout 1,

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.
Keywords: strategic sensor., hyperbolic systems, regional boundary gradient observability, Distributed systems, regional boundary reconstruction.
Mathematics Subject Classification: Primary: 93B07; Secondary: 37N3.
Citation: Adil Khazari, Ali Boutoulout. Flux reconstruction for hyperbolic systems: Sensors and simulations. Evolution Equations & 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. Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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. Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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. Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 555-566. doi: 10.3934/naco.2020055
[2]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[3]

Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 1017-1029. doi: 10.3934/dcdss.2020060
[4]

João Borges de Sousa, Bernardo Maciel, Fernando Lobo Pereira. Sensor systems on networked vehicles. Networks & Heterogeneous Media, 2009, 4 (2) : 223-247. doi: 10.3934/nhm.2009.4.223
[5]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121
[6]

Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[7]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
[8]

Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure & Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267
[9]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. A note on the one-side exact boundary controllability for quasilinear hyperbolic systems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 405-418. doi: 10.3934/cpaa.2009.8.405
[10]

Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2020, 10 (1) : 141-156. doi: 10.3934/mcrf.2019033
[11]

Tatsien Li, Libin Wang. Global exact shock reconstruction for quasilinear hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 597-609. doi: 10.3934/dcds.2006.15.597
[12]

Felipe Ponce-Vanegas. Reconstruction of the derivative of the conductivity at the boundary. Inverse Problems & Imaging, 2020, 14 (4) : 701-718. doi: 10.3934/ipi.2020032
[13]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[14]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[15]

Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control & Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1
[16]

Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations & Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079
[17]

D. J. W. Simpson. On the stability of boundary equilibria in Filippov systems. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3093-3111. doi: 10.3934/cpaa.2021097
[18]

Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157
[19]

Simon Levin, Anastasios Xepapadeas. Transboundary capital and pollution flows and the emergence of regional inequalities. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 913-922. doi: 10.3934/dcdsb.2017046
[20]

Sebastian Aniţa, Vincenzo Capasso, Ana-Maria Moşneagu. Global eradication for spatially structured populations by regional control. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2511-2533. doi: 10.3934/dcdsb.2018263

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences