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An efficient approximate method for solving delay fractional optimal control problems. (English) Zbl 1371.34016
Summary: In this paper, a new numerical method for solving the delay fractional optimal control problems (DFOCPs) with quadratic performance index is presented. In the discussed DFOCP, the fractional derivative is considered in the Caputo sense. The method is based upon the Bernoulli wavelets basis. To solve the problem, first the DFOCP is transformed into an equivalent problem with dynamical system without delay. Then, an operational matrix of Riemann-Liouville fractional integration based on Bernoulli wavelets is introduced and is utilized to reduce the problem to the solution of a system of algebraic equations. With the aid of Gauss-Legendre integration formula and Newton’s iterative method for solving a system of algebraic equations, the problem is solved approximately. Also, the convergence of the Bernoulli wavelet basis is obtained. Finally, some examples are given to demonstrate the validity and applicability of the new technique and a comparison is made with the existing results.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
34H05 Control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
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[1] Agrawal, OP, A formulation and numerical scheme for fractional optimal control problems, J. Vib. Control, 14, 1291-1299, (2008) · Zbl 1229.49045
[2] Agrawal, OP, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38, 323-337, (2004) · Zbl 1121.70019
[3] Agrawal, OP, Fractional optimal control of a distributed system using eigenfunctions, ASME. J. Comput. Nonlinear Dyn., 3, 6, (2008)
[4] Babolian, E; Fattahzadeh, F, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput., 188, 417-426, (2007) · Zbl 1117.65178
[5] Bagley, RL; Torvik, PJ, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27, 201-210, (1983) · Zbl 0515.76012
[6] Baillie, RT, Long memory processes and fractional integration in econometrics, J. Econom., 73, 5-59, (1996) · Zbl 0854.62099
[7] 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., 293, 142-156, (2015) · Zbl 1349.65504
[8] Bhrawy, AH; Ezz-Eldien, SS, A new Legendre operational technique for delay fractional optimal control problems, Calcolo, (2015) · Zbl 1341.49037
[9] Chow, TS, Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. Lett. A, 342, 148-155, (2005)
[10] Dumitru, B; Maraaba, T; Jarad, F, Fractional variational principles with delay, J. Phys. A Math. Theor., 41, 315-403, (2008) · Zbl 1152.81550
[11] El-Ajou, A; Arqub, OA; Al-Smadi, M, A general form of the generalized taylor’s formula with some applications, Appl. Math. Comput., 256, 851-859, (2015) · Zbl 1338.40007
[12] El-Ajou, A; Arqub, OA; Zhour, ZA; Momani, S, New results on fractional power series: theories and applications, Entropy, 15, 5305-5323, (2013) · Zbl 1337.26010
[13] Haddadi, N; Ordokhani, Y; Razzaghi, M, Optimal control of delay systems by using a hybrid functions approximation, J. Optim. Theory Appl., 153, 338-356, (2012) · Zbl 1251.49040
[14] Jamshidi, M; Wang, CM, A computational algorithm for large-scale nonlinear time-delay systems, IEEE Trans Syst Man Cybern, 14, 2-9, (1984) · Zbl 0538.93002
[15] Jarad, F; Abdeljawad, T; Baleanu, D, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62, 609-614, (2010) · Zbl 1209.49030
[16] Keshavarz, E; Ordokhani, Y; Razzaghi, M, A numerical solution for fractional optimal control problems via Bernoulli polynomials, J. Vib. Control, 29, 1-15, (2015) · Zbl 1373.49003
[17] Keshavarz, E; Ordokhani, Y; Razzaghi, M, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Model., 38, 6038-6051, (2014) · Zbl 1429.65170
[18] Khellat, F, Optimal control of linear time-delayed systems by linear Legendre multiwavelets, J. Optim. Theory Appl., 143, 107-121, (2009) · Zbl 1176.49037
[19] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006) · Zbl 1092.45003
[20] Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978) · Zbl 0368.46014
[21] Lancaster, P.: Theory of Matrices. Academic Press, New York (1969) · Zbl 0186.05301
[22] Li, Y, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul., 15, 2284-2292, (2010) · Zbl 1222.65087
[23] Lotfi, A; Yousefi, SA, A numerical technique for solving a class of fractional variational problems, J. Comput. Appl. Math., 237, 633-643, (2013) · Zbl 1253.65105
[24] Lotfi, A; Yousefi, SA; Dehghan, M, 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, J. Comput. Appl. Math., 250, 143-160, (2013) · Zbl 1286.49030
[25] Ma, J; Liu, J; Zhou, Z, Convergence analysis of moving finite element methods for space fractional differential equations, J. Comput. Appl. Math., 255, 661-670, (2014) · Zbl 1291.65303
[26] Magin, RL, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32, 1-104, (2004)
[27] Mainardi, F; Carpinteri, A (ed.); Mainardi, F (ed.), Fractional calculus: some basic problems in continuum and statistical mechanics, 291-348, (1997), New York · Zbl 0917.73004
[28] Marzban, HR; Razzaghi, M, Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J. Frankl. Inst., 341, 279-293, (2004) · Zbl 1070.93028
[29] Odibat, Z; Momani, S; Erturk, VS, Generalized differential transform method: application to differential equations of fractional order, Appl. Math. Comput., 197, 467-477, (2008) · Zbl 1141.65092
[30] Ozdemir, N; Agrawal, OP; Iskender, BB; Karadeniz, B, Fractional optimal control of a 2-dimensional distributed system using eigenfunctions, Nonlinear Dyn., 55, 251-260, (2009) · Zbl 1170.70397
[31] Postenko, Y, Time-fractional radial diffusion in sphere, Nonlinear Dyn., 53, 55-65, (2008) · Zbl 1170.76357
[32] Qi, H; Liu, J, Time-fractional radial diffusion in hollow geometries, Meccanica, 45, 577-583, (2010) · Zbl 1258.35120
[33] Rahimkhani, P; Ordokhani, Y; Babolian, E, Fractional-order Bernoulli wavelets and their applications, Appl. Math. Model., (2016) · Zbl 06626265
[34] Rahimkhani, P; Ordokhani, Y; Babolian, E, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl., (2016) · Zbl 06626265
[35] Rehman, M; Rahmat, RA, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16, 4163-4173, (2011) · Zbl 1222.65063
[36] Rossikhin, YA; Shitikova, MV, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50, 15-67, (1997)
[37] Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus. Springer, Berlin (2007) · Zbl 1116.00014
[38] Saeedi, H; Mohseni Moghadam, M; Mollahasani, N; Chuev, GN, A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 16, 1154-1163, (2011) · Zbl 1221.65354
[39] Safaie, E; Farahi, MH; Farmani Ardehaie, M, An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials, Comput. Appl. Math., 34, 831-846, (2015) · Zbl 1326.49047
[40] Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, Berlin (2002) · Zbl 1004.65001
[41] Tricaud, C; Chen, Y, An approximate method for numerically solving fractional order optimal control problems of general form, Comput. Math. Appl, 59, 1644-1655, (2010) · Zbl 1189.49045
[42] Wang, X.T.: Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials. Appl. Math. Comput. 184(2), 849-856 (2007) · Zbl 1114.65076
[43] Wang, Q; Chen, F; Huang, F, Maximum principle for optimal control problem of stochastic delay differential equations driven by fractional Brownian motions, Optim. Control Appl. Methods, (2014) · Zbl 1333.93266
[44] Witayakiattilerd, W, Optimal regulation of impulsive fractional differential equation with delay and application to nonlinear fractional heat equation, J. Math. Res., 5, 94-106, (2013)
[45] Yüzbasi, S, Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials, Appl. Math. Comput., 219, 6328-6343, (2013) · Zbl 1280.65075
[46] Zhu, L; Fan, Q, Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW, Commun. Nonlinear Sci. Numer. Simul., 18, 1203-1213, (2013) · Zbl 1261.35152
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