A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration

Abstract

In this paper, a new operational matrix of integration is derived using Genocchi polynomials, which is one of the Appell polynomials. By using the matrix, we develop an efficient, direct and new numerical method for solving a class of fractional optimal control problems. The fractional derivative in the dynamic constraints was replaced with the Genocchi polynomials with unknown coefficients and a Genocchi operational matrix of fractional integration. Then, the equation derived from the dynamic constraints was put into the performance index. Hence, the fractional optimal control problems will be reduced to fractional variational problems. By finding a necessary condition for the optimality for the performance index, we will obtain a system of algebraic equations that can be easily solved by using any numerical method. Hence, we obtain the value of unknown coefficients of Genocchi polynomials. Lastly, the solution of the fractional optimal control problems will be obtained. In short, the properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. The approximation approach is simple to use and computer oriented. Illustrative examples are given to show the simplicity, accuracy and applicability of the method.

Keywords

Fractional optimal control problems Genocchi operational matrix of integration numerical solution Caputo fractional derivative Riemann–Liouville fractional integration

Get full access to this article

View all access options for this article.

References

Agrawal

(2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics 38(1–4): 323–337.

Agrawal

Baleanu

(2007) A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. Journal of Vibration and Control 13(9–10): 1269–1281.

Ali

Pereira

Gama

(2015) A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems. Mathematical Methods in the Applied Sciences 39(13): 3640–3649.

Alizadeh

Effati

(2016) An iterative approach for solving fractional optimal control problems. Journal of Vibration and Control. DOI: 10.1177/1077546316633391.

Baleanu

Defterli

Agrawal

(2009) A central difference numerical scheme for fractional optimal control problems. Journal of Vibration and Control 15(4): 583–597.

Bhrawy

Doha

Baleanu

et al. (2015) An accurate numerical technique for solving fractional optimal control problems. Proceedings of The Romania Academy, Series A 16(1): 47–54.

Dehghan

Hamedi

Khosravian-Arab

(2016) A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. Journal of Vibration and Control 22(6): 1547–1559.

Doha

Bhrawy

Baleanu

et al. (2015) An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Advances in Difference Equations 2015(1): 1–17.

Erjaee

(2013) The operational matrix of fractional integration for shifted Legendre polynomials. Iranian Journal of Science and Technology (Sciences) 37(4): 439–444.

10.

Ezz-Eldien

Doha

Baleanu

et al. (2017) A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. Journal of Vibration and Control 23(1): 16–30.

11.

Frederico

Torres

(2007) Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. International Mathematical Forum 3(10): 479–493.

12.

Horadam A (1991) Genocchi polynomials. Applications of Fibonacci Numbers. In: Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer Academic, Dordrecht, pp. 145–166.

13.

Jahanshahi

Torres

(2016) A simple accurate method for solving fractional variational and optimal control problems. Journal of Optimization Theory and Applications. DOI: 10.1007/s10957-016-0884-3.

14.

Keshavarz

Ordokhani

Razzaghi

(2015) A numerical solution for fractional optimal control problems via Bernoulli polynomials. Journal of Vibration and Control 22: 3889–3903.

15.

Khader

Hendy

(2012) An efficient numerical scheme for solving fractional optimal control problems. International Journal of Nonlinear Science 14(3): 287–296.

16.

Kim

(2007) A note on the q-Genocchi numbers and polynomials. Journal of Inequalities and Applications 2007(1): 1–8.

17.

Kreyszig

(1989) Introductory Functional Analysis with Applications Vol. 81New York: Wiley.

18.

Lotfi

Dehghan

Yousefi

(2011) A numerical technique for solving fractional optimal control problems. Computers & Mathematics with Applications 62(3): 1055–1067.

19.

Lotfi

Yousefi

Dehghan

(2013) Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and Gauss quadrature rule. Journal of Computational and Applied Mathematics 250: 143–160.

20.

Mashayekhi

Razzaghi

(2016) An approximate method for solving fractional optimal control problems by hybrid functions. Journal of Vibration and Control. DOI: 10.1177/1077546316665956.

21.

Mordukhovich BS (2006) Variational Analysis and Generalized Differentiation, II: Applications [Fundamental Principles of Mathematical Sciences], vol. 331, Berlin: Springer-Verlag.

22.

Nemati

Yousefi

Soltanian

et al. (2017) An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix. Asian Journal of Control 18(6): 2272–2282.

23.

Srivastava

Garg

Choudhary

(2011) Some new families of generalized Euler and Genocchi polynomials. Taiwanese Journal of Mathematics 15(1): 283–305.

24.

Sweilam

Al-Ajami

(2015) Legendre spectral-collocation method for solving some types of fractional optimal control problems. Journal of advanced research 6(3): 393–403.

25.

Tohidi

Nik

(2015) A Bessel collocation method for solving fractional optimal control problems. Applied Mathematical Modelling 39(2): 455–465.

26.

Tohidi

Bhrawy

Erfani

(2013) A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Applied Mathematical Modelling 37(6): 4283–4294.

27.

Zahra

Hikal

(2015) Non standard finite difference method for solving variable order fractional optimal control problems. Journal of Vibration and Control. DOI: 10.1177/1077546315586646.