×

zbMATH — the first resource for mathematics

Numerical solutions for solving a class of fractional optimal control problems via fixed-point approach. (English) Zbl 1381.49031
Summary: In this paper, an optimization problem is performed to obtain an approximate solution for a class of Fractional Optimal Control Problems (FOCPs) with the initial and final conditions. The main characteristic of our approximation is to reduce the FOCP into a system of Volterra integral equations. Then, by solving this new problem, based on minimization and control the total error, we transform the original FOCP into a discrete optimization problem. By obtaining the optimal solutions of this problem, we obtain the numerical solution of the original problem. This procedure not only simplifies the problem but also speeds up the computations. The numerical solutions obtained from the proposed approximation indicate that this approach is easy to implement and accurate when applied to FOCPs.

MSC:
49M25 Discrete approximations in optimal control
49L99 Hamilton-Jacobi theories
65L03 Numerical methods for functional-differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agrawal, OP, General formulation for the numerical solution of optimal control problems, Int. J. Control, 50, 627-638, (1989) · Zbl 0679.49031
[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, A quadratic numerical scheme for fractional optimal control problems, J. Dyn. Syst. Meas. Control, 130, 011010, (2008)
[4] Agrawal, OP, A formulation and numerical scheme for fractional optimal control problems, J. Vibr. Control, 14, 1291-1299, (2008) · Zbl 1229.49045
[5] Agrawal, OP; Baleanu, D, A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vibr. Control, 13, 1269-1281, (2007) · Zbl 1182.70047
[6] Alipour, M; Rostamy, D; Baleanu, D, Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices, J. Vibr. Control, 19, 2523-2540, (2013) · Zbl 1358.93097
[7] Almeida, R., Pooseh, S., Torres, D.F.M.: Computational Methods in the Fractional Calculus of Variations. Imperial College Press, London (2015) · Zbl 1322.49001
[8] Almeida, R; Torres, DF, A discrete method to solve fractional optimal control problems, Nonlinear Dyn., 80, 1811-1816, (2014) · Zbl 1345.49022
[9] Baleanu, D; Muslih, SI; Rabei, EM, On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlinear Dyn., 53, 67-74, (2008) · Zbl 1170.70324
[10] Baleanu, D; Defterli, O; Agrawal, OP, A central difference numerical scheme for fractional optimal control problems, J. Vibr. Control, 15, 583-597, (2009) · Zbl 1272.49068
[11] Bhrawy, AH; Doha, EH; Baleanu, D; Ezz-Eldien, SS; Abdelkawy, MA, An accurate numerical technique for solving fractional optimal control problems, Differ. Equ., 15, 23, (2015) · Zbl 1370.49027
[12] Blank, L.: Numerical Treatment of Differential Equations of Fractional Order. University of Manchester, Manchester (1996)
[13] Bohannan, GW, Analog fractional order controller in temperature and motor control applications, J. Vibr. Control, 14, 1487-1498, (2008)
[14] Dehghan, M., Hamedi, E.A., Khosravian-Arab, H.: A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vibr. Control 22(6), 1547-1559 (2014) · Zbl 1365.26005
[15] Diethelm, K, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1-6, (1997) · Zbl 0890.65071
[16] Diethelm, K; Ford, NJ; Freed, AD, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29, 3-22, (2002) · Zbl 1009.65049
[17] Diethelm, K; Ford, NJ, Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248, (2002) · Zbl 1014.34003
[18] Diethelm, K; Ford, NJ, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640, (2004) · Zbl 1060.65070
[19] Doha, EH; Bhrawy, AH; Baleanu, D; Ezz-Eldien, SS; Hafez, RM, An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems, Adv. Differ. Equ., 2015, 1-17, (2015) · Zbl 1423.49018
[20] Esmaeili, S; Shamsi, M; Luchko, Y, Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials, Comput. Math. Appl., 62, 918-929, (2011) · Zbl 1228.65132
[21] Ezz-Eldien, S.S., Doha, E.H., Baleanu, D., Bhrawy, A.H.: A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vibr. Control (2015) · Zbl 1373.49029
[22] Ghandehari, MAM; Ranjbar, M, A numerical method for solving a fractional partial differential equation through converting it into an NLP problem, Comput. Math. Appl., 65, 975-982, (2013) · Zbl 1266.90181
[23] 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
[24] Hosseinpour, S., Nazemi, A.: Solving fractional optimal control problems with fixed or free final states by Haar wavelet collocation method. IMA J. Math. Control Inf. (2015) · Zbl 1397.93104
[25] Jafari, H; Seifi, S, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Nonlinear Sci. Numer. Simul., 14, 2006-2012, (2009) · Zbl 1221.65278
[26] Kamocki, R, On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput., 235, 94-104, (2014) · Zbl 1334.49010
[27] Konguetsof, A; Simos, TE, On the construction of exponentially-fitted methods for the numerical solution of the Schrödinger equation, J. Comput. Meth. Sci. Eng., 1, 143-160, (2001) · Zbl 1012.65075
[28] Kosmatove, N, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal., 70, 252-2529, (2009)
[29] Kumar, P; Agrawal, OP, An approximate method for numerical solution of fractional differential equations, Signal Process., 86, 2602-2610, (2006) · Zbl 1172.94436
[30] Li, C., Deng, W., Zhao, L.: Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations (2015). arXiv:1501.00376
[31] Li, C., Zeng, F.: Numerical Methods for Fractional Calculus, vol. 24. CRC Press, New York (2015) · Zbl 1326.65033
[32] Momani, S; Odibat, Z, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177, 488-494, (2006) · Zbl 1096.65131
[33] Momani, S; Odibat, Z, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solit. Fract., 31, 1248-1255, (2007) · Zbl 1137.65450
[34] Momani, S; Odibat, Z; Erturk, VS, Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation, Phys. Lett. A, 370, 379-387, (2007) · Zbl 1209.35066
[35] Momani, S; Odibat, Z, A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized taylor’s formula, J. Comput. Appl. Math., 220, 85-95, (2008) · Zbl 1148.65099
[36] Oldham, K.B.: The Fractional Calculus. Elsevier, Amsterdam (1974) · Zbl 0292.26011
[37] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation (2001). arXiv:math/0110241 [math.CA] · Zbl 1042.26003
[38] Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998) · Zbl 0924.34008
[39] Pooseh, S; Almeida, R; Torres, DF, Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim., 10, 363-381, (2014) · Zbl 1278.26013
[40] Rakhshan, S.A., Kamyad, A.V., Effati, S.: An efficient method to solve a fractional differential equation by using linear programming and its application to an optimal control problem. J. Vibr. Control (2015) · Zbl 1365.26008
[41] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Integrals and Derivatives of Fractional Order and Some of Their Applications. Gordon and Breach, London (1987) · Zbl 0617.26004
[42] Simos, TE; Williams, PS, On finite difference methods for the solution of the Schrödinger equation, Comput. Chem., 23, 513-554, (1999) · Zbl 0940.65082
[43] Sun, J.X.: Nonlinear Functional Analysis and its Application. Science Press, Beijin (2008)
[44] Tohidi, E; Nik, HS, A Bessel collocation method for solving fractional optimal control problems, Appl. Math. Model., 39, 455-465, (2015)
[45] 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
[46] Tricaud, C., Chen, Y.: Time-optimal control of systems with fractional dynamics. Int. J. Differ. Equ. (2010b) · Zbl 1203.49031
[47] Yousefi, SA; Lotfi, A; Dehghan, M, The use of a Legendre multi wavelet collocation method for solving the fractional optimal control problems, J. Vibr. Control, 17, 2059-2065, (2011) · Zbl 1271.65105
[48] Zamani, M; Karimi-Ghartemani, M; Sadati, N, FOPID controller design for robust performance using particle swarm optimization, Fract. Calcul. Appl. Anal., 10, 169-187, (2007) · Zbl 1141.93351
[49] Zhang, S, Positive solutions for boundary value problems of nonlinear fractional differential euations, Electron. J. Differ. Equ., 6, 1-12, (2006) · Zbl 1134.39008
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.