Exact boundary controllability of coupled hyperbolic equations

Sergei Avdonin; Abdon Choque Rivero; Luz de Teresa

International Journal of Applied Mathematics and Computer Science (2013)

  • Volume: 23, Issue: 4, page 701-710
  • ISSN: 1641-876X

Abstract

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We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.

How to cite

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Sergei Avdonin, Abdon Choque Rivero, and Luz de Teresa. "Exact boundary controllability of coupled hyperbolic equations." International Journal of Applied Mathematics and Computer Science 23.4 (2013): 701-710. <http://eudml.org/doc/262245>.

@article{SergeiAvdonin2013,
abstract = {We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.},
author = {Sergei Avdonin, Abdon Choque Rivero, Luz de Teresa},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {coupled wave equations; controllability; Riesz bases; moment problem},
language = {eng},
number = {4},
pages = {701-710},
title = {Exact boundary controllability of coupled hyperbolic equations},
url = {http://eudml.org/doc/262245},
volume = {23},
year = {2013},
}

TY - JOUR
AU - Sergei Avdonin
AU - Abdon Choque Rivero
AU - Luz de Teresa
TI - Exact boundary controllability of coupled hyperbolic equations
JO - International Journal of Applied Mathematics and Computer Science
PY - 2013
VL - 23
IS - 4
SP - 701
EP - 710
AB - We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.
LA - eng
KW - coupled wave equations; controllability; Riesz bases; moment problem
UR - http://eudml.org/doc/262245
ER -

References

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  1. Alabau-Boussouira, F. (2003). A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM Journal on Control and Optimization 42(3): 871-906. Zbl1125.93311
  2. Alabau-Boussouira, F. and Leautaud, M. (2011). Indirect controllability of locally coupled systems under geometric conditions, Comptes Rendus Mathematique 349(7-8): 395-400. Zbl1217.35113
  3. Ammar-Kohdja, A., Benabdallah, M., González-Burgos, L. and de Teresa, L. (2011). Recent results on the controllability of coupled parabolic problems: A survey, Mathematical Control and Related Fields 1(3): 267-306 Zbl1235.93041
  4. 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. Zbl0866.93001
  5. Avdonin, S.A. and Ivanov, S.A. (2001). Exponential Riesz bases of subspaces and divided differences, St Petersburg Mathematical Journal 13(3): 339-351. Zbl0999.42018
  6. Avdonin, S. and Moran, W. (2001a). Ingham-type inequalities and Riesz bases of divided differences, International Journal of Applied Mathematics and Computer Science 11(4): 803-820. Zbl1031.93098
  7. Avdonin, S. and Moran, W. (2001b). Simultaneous control problems for systems of elastic strings and beams, Systems and Control Letters 44(2): 147-155. Zbl0986.93037
  8. Avdonin, S. and Pandolfi, L. (2011). Temperature and heat flux dependence independence for heat equations with memory, in R. Sipahi, T. Vyhlidal, S.-I. Niculescu and P. Pepe (Eds.), Time Delay Systems-Methods, Applications and New Trends, Lecture Notes in Control and Information Sciences, Vol. 423, Springer-Verlag, Berlin/Heidelberg, pp. 87-101. Zbl1298.93067
  9. Biot, M. (1962). Generalized theory of acoustic propagation in porous dissipative media, The Journal of the Acoustical Society of America 34(9): 1254-1264. 
  10. Bodart, O. and Fabre, C. (1995). Controls insensitizing the norm of the solution of a semilinear heat equation, Journal of Mathematical Analysis and Applications 195(3): 658-683. Zbl0852.35070
  11. Dáger, R. (2006). Insensitizing controls for the 1-D wave equation, SIAM Journal on Control and Optimization 45(5): 1758-1768. Zbl1120.93008
  12. El Jai, A. and Hamzaoui, H. (2009). Regional observation and sensors, International Journal of Applied Mathematics and Computer Science 19(1): 5-14, DOI: 10.2478/v10006-009-0001-y. Zbl1169.93008
  13. Hansen, S. and Zuazua E. (1995). Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM Journal on Control and Optimization 33(5): 1357-1391. Zbl0853.93018
  14. Holger, S., Frehner, M. and Schmalholz S. (2010). Waves in residual-saturated porous media, in G.A. Maugin and A.V. Metrikine (Eds.), Mechanics of Generalized Continua, Advances in Mechanics and Mathematics, Vol. 21, Springer, New York, NY, pp. 179-187. 
  15. Kavian, O. and de Teresa, L. (2010). Unique continuation principle for systems of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations 16(2): 247-274. Zbl1195.35080
  16. Khapalov, A. (2010). Source localization and sensor placement in environmental monitoring, International Journal of Applied Mathematics and Computer Science 20(3): 445-458, DOI: 10.2478/v10006-010-0033-3. Zbl1211.93071
  17. Leonard, E. (1996). The matrix exponential, SIAM Review 38(3): 507-512. Zbl0858.34009
  18. Lions, J.L. (1989). Remarques préliminaires sur le contrˆole des systèmes à données incomplètes, XI Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Málaga, Spain pp. 43-54. 
  19. Najafi, M. (2001). Study of exponential stability of coupled wave systems via distributed stabilizer, International Journal of Mathematics and Mathematical Sciences 28(8): 479-491. Zbl1049.35122
  20. Najafi, M., Sarhangi G.R. and Wang H. (1997). Stabilizability of coupled wave equations in parallel under various boundary conditions, IEEE Transactions on Automatic Control 42(9): 1308-1312. Zbl0883.93044
  21. Pandolfi, L. (2009). Riesz system and the controllability of heat equations with memory Integral Equations and Operator Theory 64(3): 429-453. Zbl1187.93011
  22. Rosier, L. and de Teresa, L. (2011). Exact controllability of a cascade system of conservative equations, Comptes Rendus Mathematique 349(5): 291-296. Zbl1210.93016
  23. Russell, D. (1978). Controllability and stabilizability theory for linear partial differential equations, SIAM Review 20(4): 639-739. Zbl0397.93001
  24. Tebou, L. (2008). Locally distributed desensitizing controls for the wave equation, Comptes Rendus Mathematique 346(7): 407-412. Zbl1135.49017
  25. de Teresa, L. (2000). Insensitizing controls for a semilinear heat equation, Communications in Partial Differential Equations 25(1-2): 39-72. Zbl0942.35028
  26. Uciński, D. and Patan, M. (2010). Sensor network design for the estimation of spatially distributed processes, International Journal of Mathematics and Mathematical Sciences 20(3): 459-481, DOI: 10.2478/v10006-010-0034-2. Zbl1211.93041

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