# zbMATH — the first resource for mathematics

An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. (English) Zbl 1423.49018
Summary: In this article, we introduce a numerical technique for solving a general form of the fractional optimal control problem. Fractional derivatives are described in the Caputo sense. Using the properties of the shifted Jacobi orthonormal polynomials together with the operational matrix of fractional integrals (described in the Riemann-Liouville sense), we transform the fractional optimal control problem into an equivalent variational problem that can be reduced to a problem consisting of solving a system of algebraic equations by using the Legendre-Gauss quadrature formula with the Rayleigh-Ritz method. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.

##### MSC:
 49K15 Optimality conditions for problems involving ordinary differential equations 49M25 Discrete approximations in optimal control 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
Full Text:
##### References:
 [1] Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999) · Zbl 0924.34008 [2] Mainardi, F: Fractional Calculus Continuum Mechanics. Springer, Berlin (1997) · Zbl 0917.73004 [3] Debnath, L, A brief historical introduction to fractional calculus, Int. J. Math. Educ. Sci. Technol., 35, 487-501, (2004) [4] David, SA; Linares, JL; Pallone, EMJA, Fractional order calculus: historical apologia, basic concepts and some applications, Rev. Bras. Ensino Fis., 33, (2011) [5] Saxena, RK; Mathai, AM; Haubold, HJ, On generalized fractional kinetic equations, Physica A, 344, 657-664, (2004) [6] Lewandowski, R; Chorazyczewski, B, Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers, Comput. Struct., 88, 1-17, (2010) [7] Magin, RL: Fractional Calculus in Bioengineering. Begell House Publishers, Redding (2006) [8] Ahmad, WM; El-Khazali, R, Fractional-order dynamical models of love, Chaos Solitons Fractals, 33, 1367-1375, (2007) · Zbl 1133.91539 [9] Picozzi, S; West, B, Fractional Langevin model of memory in financial markets, Phys. Rev. E, 66, (2002) [10] Chen, W, A speculative study of $$2/3$$-order fractional Laplacian modeling of turbulence: some thoughts and conjectures, Chaos, 16, (2006) · Zbl 1146.37312 [11] Dzielinski, A; Sierociuk, D; Sarwas, G, Some applications of fractional order calculus, Bull. Pol. Acad. Sci., Tech. Sci., 58, 583-592, (2010) · Zbl 1220.80006 [12] Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, River Edge (2000) · Zbl 0998.26002 [13] Sierociuk, D; Dzielinski, A; Sarwas, G; Petras, I; Podlubny, I; Skovranek, T, Modelling heat transfer in heterogeneous media using fractional calculus, Philos. Trans. R. Soc. Lond. A, 371, (2013) · Zbl 1382.80004 [14] Srivastava, HM, Some applications of fractional calculus operators to certain classes of analytic and multivalent functions, J. Math. Anal. Appl., 122, 187-196, (1987) · Zbl 0589.30016 [15] Srivastava, HM; Aouf, MK, Some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients, Comput. Math. Appl., 30, 53-61, (1995) · Zbl 0838.30014 [16] Tarasov, VE, Fractional vector calculus and fractional maxwell’s equations, Ann. Phys., 323, 2756-2778, (2008) · Zbl 1180.78003 [17] Doha, EH; Bhrawy, AH; Ezz-Eldien, SS, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62, 2364-2373, (2011) · Zbl 1231.65126 [18] Doha, EH; Bhrawy, AH; Ezz-Eldien, SS, A new Jacobi operational matrix: an application for solving fractional differential equations, Appl. Math. Model., 36, 4931-4943, (2012) · Zbl 1252.34019 [19] Saadatmandi, A, Bernstein operational matrix of fractional derivatives and its applications, Appl. Math. Model., 38, 1365-1372, (2014) · Zbl 1427.65134 [20] Doha, EH; Bhrawy, AH; Ezz-Eldien, SS, Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method, Cent. Eur. J. Phys., 11, 1494-1503, (2013) · Zbl 1277.93034 [21] Bhrawy, AH; Zaky, MA, A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281, 876-895, (2015) · Zbl 1352.65386 [22] Bhrawy, AH, Zaky, MA, Baleanu, D: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67(2) (2015) · Zbl 1279.49020 [23] Bhrawy, AH; Doha, EH; Ezz-Eldien, SS; Gorder, RAV, A new Jacobi spectral collocation method for solving $$1+1$$ fractional Schrödinger equations and fractional coupled Schrödinger systems, Eur. Phys. J. Plus, 129, (2014) [24] Akrami, MH; Atabakzadeh, MH; Erjaee, GH, The operational matrix of fractional integration for shifted Legendre polynomials, Iran. J. Sci. Technol., Trans. A, Sci., 37, 439-444, (2013) [25] Doha, EH; Bhrawy, AH; Ezz-Eldien, SS, An efficient Legendre spectral tau matrix formulation for solving fractional sub-diffusion and reaction sub-diffusion equations, J. Comput. Nonlinear Dyn., 10, (2015) [26] Bhrawy, AH; Doha, EH; Baleanu, D; Ezz-Eldien, SS, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., (2014) · Zbl 1349.65504 [27] Bhrawy, AH; Baleanu, D; Assas, L, Efficient generalized Laguerre spectral methods for solving multi-term fractional differential equations on the half line, J. Vib. Control, 20, 973-985, (2014) · Zbl 1348.65060 [28] Djennoune, S; Bettayeb, M, Optimal synergetic control for fractional-order systems, Automatica, 49, 2243-2249, (2013) · Zbl 1364.93184 [29] Frederico, GSF; Torres, DFM, Fractional conservation laws in optimal control theory, Nonlinear Dyn., 53, 215-222, (2008) · Zbl 1170.49017 [30] Jarad, F; Abdeljawad, T; Baleanu, D, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62, 609-614, (2010) · Zbl 1209.49030 [31] Guo, TL, The necessary conditions of fractional optimal control in the sense of Caputo, J. Optim. Theory Appl., 156, 115-126, (2013) · Zbl 1263.49018 [32] Kamocki, R, On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput., 235, 94-104, (2014) · Zbl 1334.49010 [33] Dorville, R; Mophou, GM; Valmorin, VS, Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation, Comput. Math. Appl., 62, 1472-1481, (2011) · Zbl 1228.35263 [34] Tohidi, E; Nik, HS, A Bessel collocation method for solving fractional optimal control problems, Appl. Math. Model., 39, 455-465, (2015) [35] Pooseh, S; Almeida, R; Torres, DFM, A numerical scheme to solve fractional optimal control problems, Conf. Pap. Math., 2013, (2013) [36] Pooseh, S; Almeida, R; Torres, DFM, Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim., 10, 363-381, (2014) · Zbl 1278.26013 [37] Kamocki, R, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37, 1668-1686, (2014) · Zbl 1298.26023 [38] Ozdemir, N; Karadeniz, D; Iskender, BB, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373, 221-226, (2009) · Zbl 1227.49007 [39] Baleanu, D; Defterli, O; Agrawal, OMP, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15, 547-597, (2009) · Zbl 1272.49068 [40] Akbarian, T; Keyanpour, M, A new approach to the numerical solution of fractional order optimal control problems, Appl. Appl. Math., 8, 523-534, (2013) · Zbl 1279.49020
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.