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A pseudospectral method for fractional optimal control problems. (English) Zbl 1377.49019
Summary: In this article, a direct pseudospectral method based on Lagrange interpolating functions with fractional power terms is used to solve the fractional optimal control problem. As most applied fractional problems have solutions in terms of the fractional power, using appropriate characteristic nodal-based functions with suitable power leads to a more accurate pseudospectral approximation of the solution. The Lagrange interpolating functions and their fractional derivatives belong to the Müntz space; such functions are employed to show that a relationship exists between the Karush-Kuhn-Tucker conditions associated with nonlinear programming and necessary optimality conditions. Furthermore, the convergence of the method is investigated. The obtained numerical results are an indication of the behavior of the algorithm.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
93B60 Eigenvalue problems
49M37 Numerical methods based on nonlinear programming
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[1] Elnagar, G; Kazemi, M; Razzaghi, M, The pseudospectral Legendre method for discretizing optimal control problems, IEEE Trans. Autom. Control, 40, 1793-1796, (1995) · Zbl 0863.49016
[2] Fahroo, F; Ross, IM, Costate estimation by a Legendre pseudospectral method, J. Guid. Control Dyn., 24, 270-277, (2001)
[3] Benson, DA; Huntington, GT; Thorvaldsen, TP; Rao, AV, Direct trajectory optimization and costate estimation via an orthogonal collocation method, J. Guid. Control Dyn., 29, 1435-1440, (2006)
[4] Garg, D; Patterson, MA; Darby, CL; Francolin, C; Huntington, GT; Hager, WW; Rao, AV, Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a radau pseudospectral method, Comput. Optim. Appl., 49, 335-358, (2011) · Zbl 1226.49026
[5] Garg, D; Patterson, MA; Hager, WW; Rao, AV; Benson, DA; Huntington, GT, A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 49, 1843-1851, (2010) · Zbl 1219.49028
[6] Garg, D; Hager, WW; Rao, AV, Pseudospectral methods for infinite-horizon optimal control problems, Automatica, 47, 829-837, (2011) · Zbl 1215.49040
[7] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[8] Zayernouri, M; Karniadakis, G, Fractional spectral collocation method, SIAM J. Sci. Comput., 36, a40-a62, (2014) · Zbl 1294.65097
[9] Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012) · Zbl 1258.49001
[10] Almeida, R., Pooseh, S., Torres, D.F.M.: Computational Methods in the Fractional Calculus of Variations. Imperial College Press, London (2015) · Zbl 1322.49001
[11] Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. Springer Briefs in Applied Sciences and Technology. Springer, New York (2015) · Zbl 1330.49001
[12] Riewe, F, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53, 1890-1899, (1996)
[13] Agrawal, OMP, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38, 323-337, (2004) · Zbl 1121.70019
[14] Almedia, R; Torres, DFM, Leitmann’s direct method for fractional optimization problems, Appl. Math. Comput., 217, 956-962, (2010) · Zbl 1200.65049
[15] Almedia, R; Torres, DFM, A discrete method to solve fractional optimal control problems, Nonlinear Dyn., 80, 1811-1816, (2015) · Zbl 1345.49022
[16] Pooseh, S; Almeida, R; Torres, DFM, A discrete time method to the first variation of fractional order variational functionals, Cent. Eur. J. Phys., 11, 1262-1267, (2013)
[17] Baleanu, D; Defterli, O; Agrawal, OMP, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15, 583-597, (2009) · Zbl 1272.49068
[18] Tricaud, C; Chen, YQ, An approximate method for numerically solving fractional order optimal control problems of general form, Comput. Math. Appl., 59, 1644-1655, (2010) · Zbl 1189.49045
[19] Tricaud, C., Chen, Y.Q.: Solving fractional order optimal control problems in riots\(\underline{~}\)95 a general purpose optimal control problem solver. In: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey (2008)
[20] Biswas, RK; Sen, S, Free final time fractional optimal control problems, J. Frankl. Inst., 351, 941-951, (2014) · Zbl 1293.49038
[21] Yousefi, SA; Lotfi, A; Dehghan, M, The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems, J. Vib. Control, 17, 1-7, (2011) · Zbl 1271.65105
[22] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Amsterdam (2006) · Zbl 1092.45003
[23] Odibat, ZM; Shawagfeh, NT, Generalized taylor’s formula, Appl. Math. Comput., 186, 286-293, (2007) · Zbl 1122.26006
[24] Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995) · Zbl 0840.26002
[25] Borwein, P; Erdélyi, T; Zhang, J, Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Am. Math. Soc., 342, 523-542, (1994) · Zbl 0799.41015
[26] Esmaeili, S; Shamsi, M; Luchko, Y, Numerical solution of fractional differential equation with a collocation method based on Müntz polynomials, Comput. Math. Appl., 62, 918-929, (2011) · Zbl 1228.65132
[27] Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011) · Zbl 1227.65117
[28] Kang, W; Gong, Q; Ross, IM; Fahroo, F, On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems, Int. J. Robust. Nonlinear Control, 17, 1251-1277, (2007) · Zbl 1138.49027
[29] Qi Gong, I; Ross, M; Kang, W, Connection between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Comput. Optim. Appl., 41, 307-335, (2008) · Zbl 1165.49034
[30] Ruths, J., Zlotnik, A., Li Jr S.: Convergence of a Pseudospectral Method for Optimal Control of Complex Dynamical Systems. In: 50th IEEE Conference on Decision and Control. Orlando, FL, December, pp. 5553-5558 (2011)
[31] Hou, H., Hager, W. W., Rao, A. V.: Convergence of a Gauss pseudospectral method for optimal control. In: AIAA Guidance, Navigation, and Control Conference and Exhibit. American Institute of Aeronautics and Astronautics, Minnesota, vol. 8, pp. 1-9 (2012)
[32] Agrawal, OP, On a general formulation for the numerical solution of optimal control problems, Int. J. Control, 50, 627-638, (1989) · Zbl 0679.49031
[33] Elnager, G. N.: Legendre and pseudospectral Legendre approaches for solving optimal control problems. Ph.D. Thesis, Mississippi State University (1993) · Zbl 1215.49040
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