×

Solving fractional optimal control problems using Genocchi polynomials. (English) Zbl 1488.49019

Summary: In this paper, we solve a class of fractional optimal control problems in the sense of Caputo derivative using Genocchi polynomials. At first we present some properties of these polynomials and we make the Genocchi operational matrix for Caputo fractional derivatives. Then using them, we solve the problem by converting it to a system of algebraic equations. Some examples are presented to show the efficiency and accuracy of the method.

MSC:

49J21 Existence theories for optimal control problems involving relations other than differential equations
11B68 Bernoulli and Euler numbers and polynomials
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. Agoh,Convolution identities for Bernoulli and Genocchi polynomials, The Electronic Journal of Combinatorics.,21(2014), 1-65. · Zbl 1331.11013
[2] M. Alipour, and P. Allahgholi,new operatinal matrix for solving a class of optimal control problems with Jumaries modified Riemann-Liouville fractional drivative, 2017.
[3] S. Araci,Novel Identities for q-Genocchi Numbers and Polynomials, Hindawi Publishing Corporation, Journal of Function Spaces and Applications., 2012. . · Zbl 1254.11022
[4] S. Araci, M. Acikgoz, and E. Sen,Some new formulae for Genocchi numbers and polynomials involving Bernoulli and Euler polynomials, International Journal of Mathematics and Mathematical Sciences, 2014. · Zbl 1322.11020
[5] S. Araci, D. Erdal, and DJ. Kang,Some new properties on the q-Genocchi numbers and polynomials associated with q-Bernstein polynomials, Honam Mathematical Journal,33(2011), 261-270. · Zbl 1239.41009
[6] S. Araci, E. Sen, and M. Acikgoz,Theorems on Genocchi polynomials of higher order arising from Genocchi basis, Taiwanese Journal of Mathematic.,18(2014), 473-482. · Zbl 1357.11030
[7] E. Ashpazzadeh and M. Lakestani,Biorthogonal cubic Hermite spline multiwavelets on the interval for solving the fractional optimal control problems, Computational Methods for Differential Equations,4(2) (2016), 99-115. · Zbl 1424.49032
[8] E. Ashpazzadeh, M. Lakestani, and M. Razzaghi,Nonlinear constrained optimal control problems and cardinal Hermite interpolate multiscaling functions, Asian Journal of Control.,20(1) (2018), 558-567. · Zbl 1391.49060
[9] E. Ashpazzadeh, M. Lakestani, and A. Yildirim,Biorthogonal multiwavelets on the interval for solving multidimensional fractional optimal control problems with inequality constraint, Optim Control Appl Meth.,1-18(2020), Doi:10.1002/oca.2615.
[10] P. Balland and E. S¨uli,Analysis of the cell-vertex finite volume method for hyperbolic problems with variable coefficients, SIAM J. Numer. Anal.,34(1997), 1127-1151, Doi:10.1137/S0036142994264882. · Zbl 0873.65101
[11] R. K. Biswas and S. Sen,Fractional optimal control problems: a pseudo-state-space approach, Journal of Vibration and Control.,17(2011), 1034-1041. · Zbl 1271.74330
[12] T. Chiranjeevi and R. K. Biswas,Closed-form solution of optimal control problem of a fractional order system, Journal of King Saud University - Science.,3(4) (2019), 1042-1047.
[13] M. Sh. Dahaghin and H. Hassani,A new optimization method for a class of time fractional convection-diffusion-wave equations with variable coefficients, The European Physical Journal Plus., 2017. · Zbl 1380.35158
[14] N. P. Dong and H. V. Long, and A. Khastan,Optimal control of a fractional order model for granular SEIR epidemic with uncertainty, Communications in Nonlinear Science and Numerical Simulation.,88(2020). · Zbl 1454.34072
[15] SS. Ezz-Eldien, EH. Doha, AH. Bhrawy, AA. El-Kalaawy, and JA Tenreiro. Machado,A new operational approach for solving fractional variational problems depending on indefinite integrals, Communications in Nonlinear Science and Numerical Simulation.,57(2018), 246-263. · Zbl 07263284
[16] F. Ghomanjani,A numerical technique for solving fractional optimal control problems and fractional Riccati differential equations, Journal of the Egyptian Mathematical Society,24(4) (2016), 638-643. · Zbl 1352.65163
[17] R. Gorenflo, F. Mainardi, and I. Podlubny,Fractional Differential Equations, Academic Press, 8(1999), 683-699. · Zbl 0924.34008
[18] A. Haji Badali, M. S. Hashemi, and M. Ghahremani,Lie symmetry analysis for kawahara-kdv equations, Comput. Methods Differ. Equ.,1(2013), 135-145. · Zbl 1310.35020
[19] H. Hassani and E. Naragirad,A new optimization operational matrix algorithm for solving nonlinear variable-order time fractional convection-diffusion equation, 2019. · Zbl 1438.35409
[20] Y. He,Some new results on products of the ApostolGenocchi polynomials, J. Comput. Anal.Appl.,22(2017), 591-600.
[21] Y. He, S. Araci, H. M. Srivastava, and M. Acikgoz,Some new identities for the Apostol Bernoulli polynomials and the ApostolGenocchi polynomials, Applied Mathematics and Computation.,262 (2015), 31-41. · Zbl 1410.11016
[22] Y. He and T. Kim,General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials, J. Nonlinear Sci. Appl.,9(2016), 4780-4797. · Zbl 1400.11064
[23] A. Isah and Ch. Phang,New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials, Journal of King Saud University-Science.,31 (2019), 1-7.
[24] G. Jumarie,Cauchys integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order, Applied Mathematics Letters.,23(2010), 1444-1450. · Zbl 1202.30068
[25] M. EL-Kady,A Chebyshev finite difference method for solving a class of optimal control problems,International journal of computer mathematics.,80(2003), 883-895. · Zbl 1037.65065
[26] E. Keshavarz, Y. Ordokhani, and M. Razzaghi,A numerical solution for fractional optimal control problems via Bernoulli polynomials, Journal of Vibration and Control.,22(2016), 3889- 3903. · Zbl 1373.49003
[27] T. Kim,Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, arXiv preprint arXiv. 0912,4931. (2009).
[28] T. Kim, SH. Rim, DV. Dolg, and SH. Lee,Some identities of Genocchi polynomials arising from Genocchi basis, Journal of Inequalities and Applications.,43(2013). · Zbl 1284.11037
[29] E. Kreyszig,Introductory functional analysis with applications, wiley, New York,1978. · Zbl 0368.46014
[30] P. Lancaster, M. Tismenetsky,The Theory of Matrices, Academic Press. New YorkLondon xii. 1969. · Zbl 0558.15001
[31] R. J. LeVeque,Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002, Doi:10.1017/CBO9780511791253. · Zbl 1010.65040
[32] A. Lotfi and SA. Yousefi, Epsilon-Ritz method for solving a class of fractional constrained optimization problems.163(2014), 884-899. · Zbl 1386.49049
[33] A. Lotfi, SA. Yousefi, and M. Dehghan,Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule, Journal of Computational and Applied Mathematics.,250(2013), 143-160. · Zbl 1286.49030
[34] J. A. Lpez-Renteria, B. Aguirre-Hernndez, and G. Fernndez-Anaya,LMI Stability Test for Fractional Order Initialized Control Systems, Applied and Computational Methametics., An international journal,18(1) (2019), 50-61. · Zbl 1427.30007
[35] J. A. Mackenzie, T. Sonar, and E. S¨uli,Adaptive finite volume methods for hyperbolic problems, in The mathematics of finite elements and applications (Uxbridge, 1993), Wiley, Chichester, 1994, 289-297. · Zbl 0830.76072
[36] S. Mashayekhi, M. Razzaghi,An approximate method for solving fractional optimal control problems by hybrid functions, Journal of Vibration and Control.,24(2018), 1621-1631. · Zbl 1400.93122
[37] KS. Miller, B. Ross,An introduction to the fractional calculus and fractional differential equations, Wiley, 1993. · Zbl 0789.26002
[38] E. Mohammadzadeh, N. Pariz, and A. Jajarmi ,Optimal Control for a Class of Nonlinear Fractional-Order Systems Using an Extended Modal Series Method and Linear Programming Strategy, Journal of Control.,10(2016), 51-64.
[39] S. Nemati, P. M. Lima, and D. F. M. Torres,A numerical approach for solving fractional optimal control problems using modified hat functions, Communications in Nonlinear Science and Numerical Simulation,78(2019). · Zbl 1476.49033
[40] K. Rabiei, Y. Ordokhani, and E. Babolian,The Boubaker polynomials and their application to solve fractional optimal control problems, Nonlinear Dynamics.,88(2017), 1013-1026. · Zbl 1380.49058
[41] B. Ross, SG. Samko, and ER. Love,Functions that have no first order derivative might have fractional derivatives of all orders less than one, Real Analysis Exchange,20(1994), 140-157. · Zbl 0820.26002
[42] SG. Samko, AA. Kilbas, OI. Marichev, and others,Fractional integrals and derivatives, Gordon and Breach Science Publishers., Yverdon Yverdon-les-Bains, Switzerland, 1993. · Zbl 0818.26003
[43] S. Soradi-Zeid,Solving a class of fractional optimal control problems via a new efficient and accurate method, Computational Methods for Differential Equations., In Press, Doi:10.22034/cmde.2020.35875.1620. · Zbl 1449.49029
[44] V. Taherpour, M. Nazari, and A. Nemati,A new numerical Bernoulli polynomial method for solving fractional optimal control problems with vector components, Computational Methods for Differential Equations, In Press, Doi:10.22034/cmde.2020.34992.1598.
[45] B.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.