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Optimal Control of Two Linear Programming Problems

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Optimization and Applications (OPTIMA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13078))

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Abstract

On a fixed time interval, a terminal control problem generating a phase trajectory is considered. Three points are selected on the segment: two end points and one intermediate point, they correspond to the values of the trajectory. The left end of the trajectory is fixed. Finite-dimensional linear programming problems are associated with the intermediate and last moments of time, and the corresponding values of the phase trajectory should at the same time be optimal solutions of these problems. It is required to draw a phase trajectory by choosing a control so that, starting from the left end, the trajectory passes through an intermediate point and reaches the right end of the time interval. To solve the problem, a new approach is proposed based on duality theory and Lagrangian formalism. An iterative computational process of the saddle-point type is investigated. The convergence of the process in all components of the solution is proved. It is emphasized that only evidence-based computing technologies transform mathematical models into a tool for making guaranteed solutions.

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References

  1. Abdullayev, V.M., Aida-zade, K.R.: Approach to the numerical solution of optimal control problems for loaded differential equations with nonlocal conditions. Comput. Math. Math. Phys. 59(5), 696–707 (2019). https://doi.org/10.1134/S0965542519050026

    Article  MathSciNet  MATH  Google Scholar 

  2. Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Springer, Boston (1987). https://doi.org/10.1007/978-1-4615-7551-1

  3. Antipin, A.S., Khoroshilova, E.V.: Linear programming and dynamics. Trudy Inst. Mat. Mech. Ural Branch Russian Acad. Sci. 19(2), 7–25 (2015) (in Russian)

    Google Scholar 

  4. Antipin, A.S., Khoroshilova, E.V.: Linear programming and dynamics. Ural Math. J. 1(1), 3–19 (2015)

    Article  Google Scholar 

  5. Antipin, A.S., Khoroshilova, E.V.: Optimal control with connected initial and terminal conditions. Proc. Steklov Inst. Math. 289(1), 9–25 (2015). https://doi.org/10.1134/S0081543815050028

    Article  MathSciNet  MATH  Google Scholar 

  6. Antipin, A.S., Khoroshilova, E.V.: Saddle-point approach to solving problem of optimal control with fixed ends. J. Global Optim. 65(1), 3–17 (2016)

    Article  MathSciNet  Google Scholar 

  7. Antipin, A., Khoroshilova, E.: On methods of terminal control with boundary-value problems: lagrange approach. In: Goldengorin, B. (ed.) Optimization and Its Applications in Control and Data Sciences. SOIA, vol. 115, pp. 17–49. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-42056-1_2

    Chapter  MATH  Google Scholar 

  8. Antipin, A.S., Khoroshilova, E.V.: Feedback synthesis for a terminal control problem. Comput. Math. Math. Phys. 58(12), 1903–1918 (2018). https://doi.org/10.1134/S0965542518120035

    Article  MathSciNet  MATH  Google Scholar 

  9. Antipin, A.S., Khoroshilova, E.V.: Lagrangian as a tool for solving linear optimal control problems with state constraints. Optimal control and differential games. In: Proceedings of the International Conference Dedicated to the 110th Anniversary of the Birth of Lev Semenovich Pontryagin, pp. 23–26 (2018)

    Google Scholar 

  10. Antipin, A., Khoroshilova, E.: Controlled dynamic model with boundary-value problem of minimizing a sensitivity function. Optim. Lett. 13(3), 451–473 (2017). https://doi.org/10.1007/s11590-017-1216-8

    Article  MathSciNet  MATH  Google Scholar 

  11. Antipin, A.S., Khoroshilova, E.V.: Dynamics, phase constraints, and linear programming. Comput. Math. Math. Phys. 60(2), 184–202 (2020)

    Article  MathSciNet  Google Scholar 

  12. Antipin, A., Jacimovic, V., Jacimovic, M.: Dynamics and variational inequalities. Comp. Maths. Math. Phys. 57(5), 784–801 (2017)

    Article  MathSciNet  Google Scholar 

  13. Antipin, A., Vasilieva, O.: Dynamic method of Multipliers in terminal control. Comp. Maths. Math. Phys. 55(5), 766–787 (2015)

    Article  MathSciNet  Google Scholar 

  14. Dmitruk, A.V.: Convex Analysis. Moscow, MAKS-PRESS, Elementary Introductory Course (2012).(in Russian)

    Google Scholar 

  15. Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. FIZMATLIT, Moscow (2009).(in Russian)

    Google Scholar 

  16. Makarenko, A.V.: Intelligent control. In: Novikov, D.A. (ed.) Control Theory (Additional Chapters), 552p. LENAND, Moscow (2019) (in Russian)

    Google Scholar 

  17. Vasilyev, F.P.: Optimization methods. In: 2 Books. Moscow Center for Continuous Mathematical Education (2011) (in Russian)

    Google Scholar 

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Author information

Authors and Affiliations

  1. FRC, Computer Science and Control, RAS, Vavilov 40, 119333, Moscow, Russia

    Anatoly Antipin

  2. Lomonosov MSU, CMC Faculty, Leninskiye Gory, 119991, Moscow, Russia

    Elena Khoroshilova

Authors
  1. Anatoly Antipin
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  2. Elena Khoroshilova
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Editor information

Editors and Affiliations

  1. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Nicholas N. Olenev

  2. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Yuri G. Evtushenko

  3. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  4. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  5. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

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Antipin, A., Khoroshilova, E. (2021). Optimal Control of Two Linear Programming Problems. In: Olenev, N.N., Evtushenko, Y.G., Jaćimović, M., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2021. Lecture Notes in Computer Science(), vol 13078. Springer, Cham. https://doi.org/10.1007/978-3-030-91059-4_11

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  • DOI: https://doi.org/10.1007/978-3-030-91059-4_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-91058-7

  • Online ISBN: 978-3-030-91059-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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