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Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems

Received: 17 December 2016     Accepted: 30 December 2016     Published: 30 March 2017
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Abstract

In this paper, a minimum time problem for n×n co-operative hyperbolic systems involving Laplace operator and with time-delay is considered. First, the existence of a unique solution of such hyperbolic system with time-delay is proved. Then necessary conditions of a minimum time control are derived in the form of maximum principle. Finally the bang-bang principle and the approximate controllability conditions are investigated.

Published in Mathematics Letters (Volume 3, Issue 1)
DOI 10.11648/j.ml.20170301.11
Page(s) 1-11
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Time-Optimal Control Problem, Co-operative Systems, Hyperbolic Systems with Time Delay, Approximate Controllability, Bang-Bang Principle

References
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[3] J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.
[4] P. K. C. Wang, Time-optimal control of time-lag systems with time-lag control. Journal of Mathematical Analysis and Applications Vol. 52,No, 3 366- 378, (1975).
[5] G. Knowles, Time optimal control of parabolic systems with boundary condition involving time delays Journal of Optimiz.Theor. Applics,25, (1978), 563-574 .
[6] H. O. Fattorini. The Time Optimal Problem for Distributed Control of Systems Described by the Wave Equation. In: Aziz, A. K., Wingate, J. W., Balas, M. J. (eds.): Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York, San Francisco, London (1957 ).
[7] W. Krabs, On Time-Minimal Distributed Control of Vibrating Systems Governed by an Abstract Wave Equation. AppI. Math. and Optim. 13. ( 1985 ), 137-149.
[8] H. A. El-Saify, H. M. Serag and M. A. Shehata, Time-optimal control for co-operative hyperbolic systems Involving Laplace operator. Journal of Dynamical and Control systems. 15, 3, (2009), 405-423.
[9] M. A. Shehata, Some time-optimal control problems for co-operative hyperbolic systems with distributed or boundary controls. Journal of Mathematical Sciences: Advances and Applications. vol 18, No 1-2, (2012), 63-83.
[10] M. A. Shehata, Time -optimal control problem for co-operative parabolic systems with control in initial conditions, Advances in Pure Mathematics Journal , 3, No 9A, (2013), 38-43.
[11] M. A. Shehata, Dirichlet Time-Optimal Control of Co-operative Hyperbolic Systems Advanced Modeling and Optimization Journal. Volume 16, Number 2, (2014), 355-369.
[12] Byung Soo Lee, Mohammed Shehata, Salahuddin , Time -optimal control problem for co-operative parabolic systems with strong constraint control in initial conditions, Journal of Science and Technology, Vol. 4 No. 11, (2014).
[13] R. A. Adams, Sobolev Spaces. Academic Prees, New York. (1975).
[14] J. Fleckinger, J. Herna'ndez and F. DE. The'lin, On the existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems. Rev. R. Acad. Cien. Serie A. Mat. 97, 2 (2003), 461-466.
[15] A. Friedman, Optimal control for parabolic variational inequalities. SIAM Journal of Control and Optimization , 25, 482-497, (1987).
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  • APA Style

    Hussein El-Saify, Mohammed Shehata. (2017). Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems. Mathematics Letters, 3(1), 1-11. https://doi.org/10.11648/j.ml.20170301.11

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    ACS Style

    Hussein El-Saify; Mohammed Shehata. Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems. Math. Lett. 2017, 3(1), 1-11. doi: 10.11648/j.ml.20170301.11

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    AMA Style

    Hussein El-Saify, Mohammed Shehata. Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems. Math Lett. 2017;3(1):1-11. doi: 10.11648/j.ml.20170301.11

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  • @article{10.11648/j.ml.20170301.11,
      author = {Hussein El-Saify and Mohammed Shehata},
      title = {Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems},
      journal = {Mathematics Letters},
      volume = {3},
      number = {1},
      pages = {1-11},
      doi = {10.11648/j.ml.20170301.11},
      url = {https://doi.org/10.11648/j.ml.20170301.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20170301.11},
      abstract = {In this paper, a minimum time problem for n×n co-operative hyperbolic systems involving Laplace operator and with time-delay is considered. First, the existence of a unique solution of such hyperbolic system with time-delay is proved. Then necessary conditions of a minimum time control are derived in the form of maximum principle. Finally the bang-bang principle and the approximate controllability conditions are investigated.},
     year = {2017}
    }
    

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    T1  - Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems
    AU  - Hussein El-Saify
    AU  - Mohammed Shehata
    Y1  - 2017/03/30
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    N1  - https://doi.org/10.11648/j.ml.20170301.11
    DO  - 10.11648/j.ml.20170301.11
    T2  - Mathematics Letters
    JF  - Mathematics Letters
    JO  - Mathematics Letters
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    EP  - 11
    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ml.20170301.11
    AB  - In this paper, a minimum time problem for n×n co-operative hyperbolic systems involving Laplace operator and with time-delay is considered. First, the existence of a unique solution of such hyperbolic system with time-delay is proved. Then necessary conditions of a minimum time control are derived in the form of maximum principle. Finally the bang-bang principle and the approximate controllability conditions are investigated.
    VL  - 3
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

  • Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia

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