A computational method on derivative variations of optimal control

Volume 28, Issue 2, pp 203--212 http://dx.doi.org/10.22436/jmcs.028.02.08

Publication Date: May 27, 2022 Submission Date: February 08, 2022 Revision Date: March 02, 2022 Accteptance Date: April 16, 2022

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Authors

E. Rentsen - The Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, National University of Mongolia, Mongolia. M. Kamada - Department of Computer and Information Sciences, Ibaraki University, Japan. A. Radwan - Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Al-Jouf-Sakaka, Kingdom of Saudi Arabia. - Department of Mathematics, Faculty of Science, Sohag University, Egypt. W. Alrashdan - Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Al-Jouf-Sakaka, Kingdom of Saudi Arabia.

Abstract

In this paper, we propose an algorithm for solving optimal control problems in a class of continuously differentiable control functions with bounded derivatives. Based on derivative variations [R. Enkhbat, B. Barsbold, J. Mongolian Math. Soc., \(\bf 17\) (2013), 27--39], we derive new optimality conditions for the original problem. An algorithm has been constructed based on the optimality conditions. The convergence of the proposed algorithm has been proved. The algorithm was tested on some well known numerical examples.

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Keywords

• Calculus of variations
• optimal control
• derivative variation
• optimality conditions

MSC

•  49M05
•  49K99

References

• [1] J. T. Betts, Practical methods for optimal control using nonlinear programming, Soc. Ind. Appl. Math., Philadelphia, PA (2001)
  • View Article
  • MathSciNet


• [2] R. Bulirsch, E. Nerz, H. J. Pesch, O. von Stryk, Combining direct and indirect methods in optimal control: range maximization of a hang glider, Optimal Control, Birkhauser, Basel (1993)
  • View Article
  • MathSciNet


• [3] F. L. Chernous'ko, A. A. Lyubushin, Method of successive approximations for solution of optimal control problems, Optimal Control Appl. Methods, 3 (1982), 101--114
  • View Article
  • MathSciNet


• [4] T. M. El-Gindy, H. M. El-Hawary, M. S. Salim, M. El-Kady, A Chebyshev approximation for solving optimal control problems, Comput. Math. Appl.,, 29 (1995), 35--45
  • View Article
  • MathSciNet


• [5] R. P. Enid, Optimal control and the calculus of variations, University Press, Oxford (1995)
  • View Article


• [6] R. Enkhbat, B. Barsbold, Derivative variation approach for some class of optimal control, J. Mongolian Math. Soc., 17 (2013), 27--39
  • MathSciNet


• [7] W. Fleming, R. Rishwl, Deterministic and stochastic optimal control, Springer, New York (1975)
  • View Article
  • MathSciNet


• [8] G. S. Frederico, P. Giordano, A. A. Bryzgalov, M. J. Lazo, Calculus of variations and optimal control for generalized functions, Nonlinear Anal., 216 (2022), 46 pages
  • View Article
  • MathSciNet


• [9] D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson, G. T. Huntington, A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46 (2010), 1843--1851
  • View Article
  • MathSciNet


• [10] P. E. Gill, W. Murray, M. A. Saunders,, User’s guide for SNOPT 5.3: a Fortran package for large-scale nonlinear programming, Report NA97-5, University of California, , San Diego (1977)
  • View Article


• [11] A. B. Modelon, JModelica.org , User Guide 1.6.0, ()
  • View Article


• [12] H. S. NiK, S. Effati, M. Shirazian, An approximate analytical solution for the Hamilton-Jacobi-Bellman equation via Homotopyperturbation method, Appl. Math. Modeling,, 36 (2012), 5614--5623
  • View Article


• [13] A. Radwan, H. L. To, W. J. Hwang, Optimal control for bufferbloat queue management using indirect method with parametric optimization, Scientific Prog.,, 2016 (2016), 10 pages
  • View Article


• [14] A. Radwan, O. Vasilieva, R. Enkhbat, A. Griewank, J. Guddat, Parametric approach to optimal control, Optimization Lett., 6 (2012), 1303--1316
  • View Article
  • MathSciNet


• [15] Z. Rafiei, B. Kafash, S. M. Karbassi, A new approach based on using Chebyshev wavelets for solving various optimal control problems, Comput. Appl. Math., 37 (2018), 144--157
  • View Article
  • MathSciNet


• [16] A. V. Rao, A survey of numerical methods for optimal control, Adv. Astronautical Sci.,, 135 (2009), 497--528
  • View Article
  • MathSciNet


• [17] K. L. Teo, B. Li, C. Yu, V. Rehbock, Applied and computational optimal control, Springer, Switzerland (2021)
  • View Article
  • MathSciNet


• [18] O. V. Vasilieva, Optimization methods, World Fedration Publishers Company, Atlanta (1996)
  • View Article


• [19] O. Vasilieva, Maximum principle and its extension for bounded control problems with boundary conditions, Int. J. Math. Math. Sci., 2004 (2004), 1855--1879
  • View Article
  • MathSciNet


• [20] O. Vasilieva, Interior variations in dynamics optimization problems, Optimization, 57 (2008), 807--825
  • View Article
  • MathSciNet


• [21] O. Vasilieva, Interior variations method for solving optimal control problems, Optimization,, 58 (2009), 897--916
  • View Article
  • MathSciNet


• [22] L. E. Zabello, On the theory of necessary conditions for optimality with delay and derivative of the control, Differ. Uravn.,, 25 (1989), 371--379
  • View Article
  • MathSciNet