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A numerical technique for solving fractional optimal control problems. (English) Zbl 1228.65109
Summary: We present a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

##### MSC:
 65L05 Numerical methods for initial value problems 49M30 Other numerical methods in calculus of variations (MSC2010) 34A08 Fractional ordinary differential equations and fractional differential inclusions 26A33 Fractional derivatives and integrals 45J05 Integro-ordinary differential equations 49K21 Optimality conditions for problems involving relations other than differential equations
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##### References:
 [1] Bagley, R.L.; Torvik, P.J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. rheol., 27, 201-210, (1983) · Zbl 0515.76012 [2] Bagley, R.L.; Torvik, P.J., Fractional calculus in the transient analysis of viscoelastically damped structures, Aiaa j., 23, 918-925, (1985) · Zbl 0562.73071 [3] Magin, R.L., Fractional calculus in bioengineering, Crit. rev. biomed. eng., 32, 1-104, (2004) [4] Chow, T.S., Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. lett. A, 342, 148-155, (2005) [5] Zamani, M.; Karimi-Ghartemani, M.; Sadati, N., FOPID controller design for robust performance using particle swarm optimization, J. fract. calc. appl. anal., 10, 169-188, (2007) · Zbl 1141.93351 [6] Agrawal, O.P., A general formulation and solution scheme for fractional optimal control problems, Nonlinear dynam., 38, 323-337, (2004) · Zbl 1121.70019 [7] Agrawal, O.P.; Baleanu, D., A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. vib. control, 13, 1269-1281, (2007) · Zbl 1182.70047 [8] Tricaud, C.; Chen, Y.Q., An approximation method for numerically solving fractional order optimal control problems of general form, Comput. math. appl., 59, 1644-1655, (2010) · Zbl 1189.49045 [9] Agrawal, O.P., A formulation and numerical scheme for fractional optimal control problems, J. vib. control, 14, 1291-1299, (2008) · Zbl 1229.49045 [10] Agrawal, O.P., A quadratic numerical scheme for fractional optimal control problems, Trans. ASME, J. dyn. syst. meas. control, 130, 1, 011010-1-011010-6, (2008) [11] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. methods partial differential equations, 26, 448-479, (2010) · Zbl 1185.65187 [12] Dehghan, M.; Manafian, J.; Saadatmandi, A., The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. naturforsch., 65a, 935-949, (2010) [13] Dehghan, M.; Yousefi, S.A.; Lotfi, A., The use of he’s variational iteration method for solving the telegraph and fractional telegraph equations, Int. J. numer. methods biomed. eng., 27, 219-231, (2011) · Zbl 1210.65173 [14] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput. math. appl., 59, 1326-1336, (2010) · Zbl 1189.65151 [15] A. Saadatmandi, M. Dehghan, A Legendre collocation method for fractional integro-differential equations, J. Vib. Control, in press (doi:10.1177/1077546310395977). · Zbl 1271.65157 [16] Wang, J.R.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear anal. RWA, 12, 262-272, (2011) · Zbl 1214.34010 [17] Rabei, E.M.; Nawafleh, K.I.; Hijjawi, R.S.; Muslih, S.I.; Baleanu, D., The Hamilton formalism with fractional derivatives, J. math. anal. appl., 327, 891-897, (2007) · Zbl 1104.70012 [18] Baleanu, D., About fractional quantization and fractional variational principles, Commun. nonlinear sci. numer. simul., 14, 2520-2523, (2009) [19] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and application of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003 [20] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives theory and applications, (1993), Gordon and Breach New York · Zbl 0818.26003 [21] Rabei, E.M.; Almayteh, I.; Muslih, S.I.; Baleanu, D., Hamilton – jacobi formulation of systems within caputo’s fractional derivative, Phys. scr., 77, (2008), Article Number 015101 · Zbl 1145.70011 [22] Tarasov, V.E., Fractional vector calculus and fractional maxwell’s equations, J. ann. physics, 323, 2756-2778, (2008) · Zbl 1180.78003 [23] Gelfand, I.M.; Fomin, S.V., Calculus of variation (R.A. silverman, trans.), (1963), PrenticeHall [24] Kreyszig, E., Introductory functional analysis with applications, (1978), John Wiley and sons, Inc. · Zbl 0368.46014 [25] Rivlin, T.J., An introduction to the approximation of functions, (1981), Dover Publications · Zbl 0189.06601
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