New class of hybrid explicit methods for numerical solution of optimal control problems

Document Type : Research Article

Authors

• M. Ebadi ¹
• I. Malih Maleki ²
• A. Ebadian ³

¹ Department of Mathematics, University of Farhangian, Tehran, Iran.

² Department of Mathematics, Payam-e-Nour University, Tehran, Iran.

³ Department of Mathematics, Urmia university, Urmia, Iran.

10.22067/ijnao.2021.67961.1005

Abstract

Forward-backward sweep method (FBSM) is an indirect numerical method used for solving optimal control problems, in which the differential equation arising from this method is solved by the Pontryagin’s maximum principle. In this paper, a set of hybrid methods based on explicit 6th-order RungeKutta method is presented for the FBSM solution of optimal control problems. Order of truncation error, stability region, and numerical results of the new hybrid methods were compared with those of the 6th-order Runge Kutta method. Numerical results show that new hybrid methods are more accurate than the 6th-order Runge–Kutta method and that their stability regions are also wider than that of the 6th-order Runge–Kutta method.

Keywords

• FBSM
• OCP
• Stability analysis
• Hybrid methods

20.1001.1.24236977.2021.11.2.3.2

Main Subjects

• Numerical methods for differential equations and its applications

---

References

 1. Ameen, I., Hidan, M., Mostefaoui, Z., and Ali, H.M. An efficient algorithms for solving the fractional optimal control of SIRV epidemic model with a combination of vaccination and treatment, Chaos Solitons Frac tals, 137 (2020), 109892, 12 pp.

2. Bhih, A.E., Ghazzali, R., Rhila, S.B., Rachik, M., and Laaroussi, A.E.A. A discrete mathematical modeling and optimal control of the rumor propagation in online social network, Discrete Dyn. Nat. Soc. 2020, Art. ID 4386476, 12 pp.
3. Duran, M.Q., Candelo, J.E., and Ortiz, J.S. A modified backward/forward sweep-based method for reconfiguration of unbalanced distribution networks, Int. J. Electr. Comput. Eng. (IJECE), 1(9) (2019), 85–101.
4. Ebadi, M. Hybrid BDF methods for the numerical solution of ordinary differential equation, Numer. Algorithms, 55(2010), 1–17.
5. Ebadi, M. A class of multi-step methods based on a super-future points technique for solving IVPs, Comput. Math. Appl. 61(2011), 3288–3297.
6. Ebadi, M. Class 2+1 hybrid BDF-like methods for the numerical solution of ordinary differential equation, Calcolo, 48(2011), 273–291.
7. Ebadi, M. New class of hybrid BDF methods for the computation of numerical solution of IVPs, Numer. Algorithms, 79(2018), 179–193.
8. Ebadi, M., Haghighi, A.R., Malih maleki, I., and Ebadian, A. FBSM solution of optimal control problems using hybrid Runge–Kutta based meth ods, J. Math. Ext. 15(4) (2021), In press.
9. Ebadi, M., Malih maleki, I., Haghighi, A.R., and Ebadian, A. An explicit single-step method for numerical solution of optimal control problems, Int. J. Ind. Math. 13(1) (2021), 71–89.
10. Jain, M.K. Numerical solution of differential equations, 2nd Edition, New Age International publishers, 2002.
11. Kheiri, H. and Jafari, M. Optimal control of a fractional-order model for the HIV/AIDS epidemic, Int. J. Biomath. 11(7) (2018), 1850086, 23 pp.
12. Kongjeen, Y., Bhumkittipich, K., Mithulananthan, N., Amiri, I. S., and Yupapin, P. A modified backward and forward sweep method for microgrid load flow analysis under different electric vehicle load mathematical models, Electr. Pow. Syst. Res., 168(2019), 46–54.
13. Lenhart, S. and Workman, J.T. Optimal control applied to biological models, Chapma &Hall/CRC, Boca Raton, 2007.
14. Lewis, F.L., Vrabie, D.L., and Syrmos, W.L. Optimal Control, 2ed, John Wiley, sons Inc, 1995.
15. Lhous, M., Rachik, M., Laarabi, H., and Abdelhak, A. Discrete mathematical modeling and optimal control of the marital status: the monog amous marriage case, Adv. Difference Equ. 2017, Paper No. 339, 16 pp.

16. McAsey, M., Moua, L., and Han, W. Convergence of the forward backward sweep method in optimal control, Comput. Optim. Appl., 53(2012), 207–226.
17. Moualeu, D.P., Weiser, M., Ehrig, R., and Deuflhard, P. Optimal control for tuberculosis model with undetected cases in Cameroon, Commun. Nonlinear Sci. Numer. Simul., 20(2015), 986–1003.
18. Rafiei, Z., Kafash, B., and Karbassi, S.M. A computational method for solving optimal control problems and their applications, Control and Optimization in Applied Mathematics, 2(1), (2017), 1–13.
19. Rodrigues, H.S., Teresa, M., Monteiro, T., and Torres, D.F.M. Optimalcontrol and numerical software: an overview, Systems Theory: Perspectives, Applications and Developments , Nova Science publishers, 2014.
20. Rose, G.R. Numerical methods for solving optimal control problems, University of Tennesse, Konxville, A Thesis for the master of science Degree, 2015.
21. Saleem, R., Habib, M., and Manaf, A. Review of forward-backward sweep method for unbounded control problem with payoff term, Sci. Int., 27(1) (2014), 69–72.
22. Sana, M., Saleem, R., Manaf, A., and Habib, M. Varying forward backward sweep method using Runge–Kutta, Euler and Trapezoidal scheme as applied to optimal control problems, Sci. Int. , 27(2015), 839–843.
23. Silveira, M., Nascimento, J.C., Marques, J.S., Marçal, A.R.S. Compari son of segmentation methods for melanoma diagnosis in Dermoscopy images, IEEE Journal of Selected Topics in Signal Processing, 3(1) (2009), 35–45.