On the numerical solution of optimal control problems via Bell polynomials basis

Document Type : Research Article

Authors

• M.R. Dadashi ¹
• A.R. Haghighi ²
• F. Soltanian ¹
• A. Yari ¹

¹ Department of Mathematics, Payame Noor University, Tehran, Iran.

² Department of Mathematics, Technical and Vocational University, Tehran, Iran.

10.22067/ijnao.v10i2.86884

Abstract

We present a new numerical approach to solve the optimal control problems (OCPs) with a quadratic performance index. Our method is based on the Bell polynomials basis. The properties of Bell polynomials are explained. We also introduce the operational matrix of derivative for Bell polynomials. The chief feature of this matrix is reducing the OCPs to an optimization problem. Finally, we discuss the convergence of the new technique and present some illustrative examples to show the effectiveness and applicability of the proposed scheme. Comparison of the proposed method with other previous methods shows that this method is accurate.

Keywords

• Optimal control problems
• Bell polynomial
• Best approximation
• Operational matrix of derivative

20.1001.1.24236977.2020.10.2.10.2

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References

1. Aguilar, C. and Krener, A. Numerical solutions to the Bellman equation of optimal control, J. Optim. Theory Appl. 160 (2014) 527–552.

2. Ahmed, H.F. and Melad, M.B. A new approach for solving fractional optimal control problems using shifted ultraspherical polynomials, Prog. Fract. Differ. Appl. 4(3) (2018) 179–195.

3. Akbarian, T. and Keyanpour, M. A new approach to the numerical solution of FOCPs, Applications and Applied Mathematics, 8 (2) (2013) 523–534.

4. Bell, E.T. Exponential polynomials. Ann. Math. (2) 35 (1934), no. 2, 258–277.

5. Boyadzhiev, K.N. Exponential polynomials, Stirling numbers and evaluation of some gamma integrals, Abstr. Appl. Anal. 2009, Art. ID 168672, 18 pp.

6. Feng, Q. and Guo B.N. Relations, among Bell polynomials, central factorial numbers, and central Bell polynomials, Mathematical Sciences and Applications, 7 (2) (2019) 191–194.

7. Frego, M. Numerical methods for optimal control problems with application to autonomous vehicles, Ph.D. Thesis, University of Trento, 2014.

8. Ghomanjani, F. and Farahi, M.H. Optimal control of switched systems based on Bezier control points, Int. J. Intell. Syst. Appl. 7 (2012) 16–22.

9. Grigoryev, I., Mustafina, S. and Larin, O. Numerical solution of optimal control problems by the method of successive approximations, Int. J. Pure Appl. Math. 112(3) (2017) 599–604.

10. Inman, D.J. Vibration with control, John Wiley Sons, Ltd. 2006.

11. Kafash, B., Delavarkhalafi, A., Karbassi, M. and Boubaker, K. A numerical approach for solving optimal control problems using the Boubaker polynomials expansion scheme, J. Interpolat. Approx. Sci. Comput. 2014, Art. ID 00033, 18 pp.

12. Kreyszig, E. Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978.

13. Lancaster, P. Theory of Matrices, New York, Academic Press, 1969.

14. Lewis, F.L., Vrabie, D.L. and Syrmos, V.L. Optimal control, Third edition. John Wiley & Sons, Inc., Hoboken, NJ, 2012.

15. Mirzaee, F. Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polynomials, Comput. Methods Differ. Equ. 5(2) (2017) 88–102.

16. Oruh, I.B. and Agwu, U.E. Application of Pontryagin’s maximum principles and Runge-Kutta methods in optimal control problems, IOSR Journal of mathematics, 11(5) (2015) 43–63.

17. Pesch, H.J. A practical guide to the solution of real life optimal control problems Parametric optimization, Control Cybernet. 23 (1994) 7–60.,

18. Ramazani, M. Numerical solution of optimal control problems by using a new second kind Chebyshev wavelet, Comput. Methods Differ. Equ. 4 (2016),, 4(2) (2016) 162–169.

19. Rogalsky, T. Bezier parameterization for optimal control by differential evolution, Proceedings of the 14th annual conference companion on Genetic and evolutionary computation, (2012) 523–530.

20. Rose, G.R. Numerical methods for solving optimal control problems, Master’s Thesis, University of Tennessee, 2015.

21. Sharif, H.R.,Vali, M.A., Samava M. and Gharavisi, A.A. A new algorithm for optimal control of time-delay systems, Appl. Math. Sci. (Ruse) 5(12) (2011) 595–606.

22. Stanley, R.P. Enumerative combinatorics, Cambridge University Press, 2011.

23. Wakhare, T. Refinements of the Bell Stirling numbers, Trans. Comb. 7(4) (2018) 25–42.

24. Yari, A.A. and Mirnia, M. Solving optimal control problems by using hermite polynomials, Comput. Methods Differ. Equ. 8(2) (2020) 314–329.

25. Yari, A.A., Mirnia M. and Lakestani, M. Investigation of optimal control problems and solving them by using Bezier polynomials, Appl. Comput. Math., 16(2) (2017) 133–147.

26. Yousefi, S.A., Lotfi, A. and Dehghan, M. The use of a Legendre multi wavelet collocation method for solving the fractional optimal control problems, J. Vib. Control 17(13) (2011) 2059–2065.