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Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints

Received: 6 September 2015     Accepted: 17 September 2015     Published: 12 October 2016
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Abstract

In this paper, the minimum time problem for differential systems of parabolic type with distributed control and control - state constraints are considered. The minimum time problem is replaced by an equivalent one with fixed time and the necessary optimality conditions of time-optimal control are obtained by using the generalized Dubovitskii-Milyutin Theorem (see [1]).

Published in Mathematical Modelling and Applications (Volume 1, Issue 1)
DOI 10.11648/j.mma.20160101.11
Page(s) 1-7
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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.

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Copyright © The Author(s), 2016. Published by Science Publishing Group

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Keywords

Time-Optimal Control Problem, Parabolic System, Dubovitskii - Milyutin Method, Canonical Approximations, Optimality Conditions

References
[1] W. Kotarski, Characterization of Pareto optimal points in problems with multi equality constraints, Optimization 20 (1989), 93–106.
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[4] R.A. Adams, Sobolev Spaces, Academic Prees, New York, (1975).
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[6] Lions, J. L. (1971). Optimal control of systems governed by partial differential equations. Band: Springer-verlag.
[7] Lions, J. L., & Magenes, E. (1972). Nonhomogeneous boundary value problem and applications. New York: Spring-Verlage.
[8] D.G.Luenberger, Optimization by vector space methods. New York: Wiley. (1969).
[9] W. Kotarski, Some problems of optimal and pareto optimal control for distributed parameter systems. Katowice, Poland: Reports of Silesian University, 1668. (1997).
[10] Ahmed Zabel & Maryam Alghamdi(2011). Garding’s Inequality for Elliptic Differential Operator with Infinite Number of Variables. International Journal of Mathematics and Mathematical SciencesVol 2011,-10.
[11] Fattorini, H. O. (1998). Infinite dimensional optimization theory and opotimal control. Cambridge: Cambridge University Press
[12] Wang, P. K. C. (1975). Optimal control of parabolic systems with boundary conditions involving time delays. SIAM Journal of Control and Optimization,13, 274-293.
[13] Wong, K. H. (1987). Optimal control computation for parabolic systems with boundary conditions involving time delays. Journal of Optimization Theory and Applications,53, 475-507.
[14] El-Saify, H. A., Serag. H. M., & Shehata, M. A. (2009). Time-optimal control for co-operative hyperbolic systems involving Laplace operator. Journal of Dynamical and Control systems 15(3), 405-423.
[15] El-Saify, H.A., Shehata,M.A. Time-Optimal Control of Petrowsky Systems with Infinitely Many Variables and Control-State Constraints Studies in Mathematical Sciences Vol. 2, No. 1, 2011, pp. 01-15.
[16] Kotarski, W., El-Saify, H. A., & Shehata, M. A. (2007). Time optimal control of parabolic lag system with infinite variables. Journal of the Egyptian mathematical society,15(1), 21-34.
[17] Mohammed 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.
[18] Mohammed 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.
[19] Mohammed Shehata, Dirichlet Time-Optimal Control of Co-operative Hyperbolic Systems Advanced Modeling and Optimization Journal. Volume 16, Number 2, (2014),355-369.
[20] 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).
[21] Mohammed Shehata, Time-optimal control of 4-th order systems.Proceedings of The 4-th International Conference on Computer Science and Computational Math ematics (ICCSCM 2015), 250-255.
[22] Mohammed Shehata, Abo-el-nour N. Abd-alla, M.A. Bakhit Time-optimal control of parabolic systems involving operator of 4th order. SYLWAN Journal.Vol. 159,Issue. 8 (2015).
[23] Mohammed Shehata, Time-optimal Neumann boundary Control of 4th Order Systems. CIENCIA E TECNICA Journal.Vol. 30, (2015).
[24] H. A. El-Saify and M. A. Shehata. Time optimal control problem for parabolic systems involving different types of operators. Journal of The Egyptian Mathematical Society. Vol 17 (2009).
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    Mohammed Shehata. (2016). Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints. Mathematical Modelling and Applications, 1(1), 1-7. https://doi.org/10.11648/j.mma.20160101.11

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    ACS Style

    Mohammed Shehata. Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints. Math. Model. Appl. 2016, 1(1), 1-7. doi: 10.11648/j.mma.20160101.11

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    AMA Style

    Mohammed Shehata. Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints. Math Model Appl. 2016;1(1):1-7. doi: 10.11648/j.mma.20160101.11

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  • @article{10.11648/j.mma.20160101.11,
      author = {Mohammed Shehata},
      title = {Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints},
      journal = {Mathematical Modelling and Applications},
      volume = {1},
      number = {1},
      pages = {1-7},
      doi = {10.11648/j.mma.20160101.11},
      url = {https://doi.org/10.11648/j.mma.20160101.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20160101.11},
      abstract = {In this paper, the minimum time problem for differential systems of parabolic type with distributed control and control - state constraints are considered. The minimum time problem is replaced by an equivalent one with fixed time and the necessary optimality conditions of time-optimal control are obtained by using the generalized Dubovitskii-Milyutin Theorem (see [1]).},
     year = {2016}
    }
    

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    JO  - Mathematical Modelling and Applications
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    AB  - In this paper, the minimum time problem for differential systems of parabolic type with distributed control and control - state constraints are considered. The minimum time problem is replaced by an equivalent one with fixed time and the necessary optimality conditions of time-optimal control are obtained by using the generalized Dubovitskii-Milyutin Theorem (see [1]).
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Author Information
  • Department of Mathematics, Faculty of Science, Jazan University, Kingdom of Saudi Arabia

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