Document Type : Research Paper
Author
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.
Abstract
In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature formula to reduce the fractional optimal control problem to solving a system of nonlinear algebraic equations that greatly simplifies the problem. Finally, some examples are included to confirm the efficiency and accuracy of the proposed method.
Keywords
[1] O.M.P. Agrawal, A general formulation and solution scheme for fractional optimal control problem, Nonlinear Dynam., 38 (2004) 323-337.
[2] O.M.P. Agrawal, A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vib. Control, (2007) 1269-1281.
[3] O.M.P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, J. Vib. Control, 14 (2008) 1291-1299.
[4] R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983) 201--210.
[5] R.L. Bagley and P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23 (1985) 918-925.
[6] D. Baleanu, O. Defterli, and O.M.P. Agrawal, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15 (2009) 547-597.
[7] G. Bohannan, Analog fractional order controller in temperature and motor control applications, J. Vibr. Control, 14 (2008) 1487-1498.
[8] K.B. Datta and B.M. Mohan, Orthogonal Functions in Systems and Control, World Scientific, Singapore, 1995.
[9] R.A. Devore and L.R. Scott, Error bounds for Gaussian quadrature and weighted-L1 polynomial approximation, SIAM J. Numer. Anal., 21 (1984) 400-412.
[10] K. Diethelm and N.J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004) 621--640.
[11] W. Grzesikiewicz, A. Wakulicz, and A. Zbiciak, Non-linear problems of fractional in modelling of mechanical systems, Int. J. Mech. Sci., 70 (2013) 90-89.
[12] J.H. He, Some applications of nonlinear fractional differential equations and their applications, Bull. Sci. Technol., 15 (1999) 86-90.
[13] M. Ichise,Y. Nagayanagi, and T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electroanal. Chem., 33 (1971) 253-265.
[14] H. Jafari, and H. Tajadodi, Fractional order optimal control problems via the operational matrices of Bernstein polynomials, U.P.B. Sci. Bull., Series A, 76(3) (2014) 115-128.
[15] Y. Jiang, X. Wang, and Y. Wang, On a stochastic heat equation with first order fractional noises and applications to finance, J. Math. Anal, Appl., 396 (2012) 656-669.
[16] E. Keshavarz, Y. Ordokhani, and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, J. Vib. Control, 29 (2015) 1-15.
[17] R. Lewandowski, and B. Chorazyczewski, Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers, Comput. Struct., 88 (2010) 1-17.
[18] A. Lotfi, M. Dehghan, and S.A. Yousefi, A numerical technique for solving fractional optimal control problems, Comput. Math. Appl., 62 (2011) 1055-1067.
[19] A. Lotfi, S.A. Yousefi, and Mehdi Dehghanb, 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 (2013) 143-160
[20] R.L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004) 1-104.
[21] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fract. Fract. Calculus Contin. Mech., 387 (1997) 291--348.
[22] Y.A. Rossikhin and M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997) 15-67.
[23] N. Sebaa, Z.E.A. Fellah, W. Lauriks, and C. Depollier, Application of fractional calculus to ultrasonic wave propagation in human cancellous bone, Signal Process., 86 (2006) 2668-2677.
[24] V.E. Tarasov, Fractional vector calculus and fractional Maxwells equations, Annals of Physics, 323 (2008) 2756-2778.
[25] J.A. Tenreiro Machado, P. Stefanescu, O. Tintareanu, and D. Baleanu, Fractional calculus analysis of the cosmic microwave background, Rom. Rep. Phys., 65 (2013) 316-323.
[26] E. Tohidi and H. Saberi Nik, A Bessel collocation method for solving fractional optimal control problems, Appl. Math. Model., 39(2) (2015) 455-465.
[27] C. Tricaud and Y.Q. Chen, An approximation method for numerically solving fractional order optimal control problems of general form, Comput. Math. Appl., 59 (2010) 1644-1655.
[28] S.A. Yousefi, A. Lotfi, and M. Dehghan, The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems, J. Vib. Control, 13 (2011) 1-7.
[29] S.B. Yuste, L. Acedo, and K. Lindenberg, Reaction front in an A + B → C reaction-subdiffusion process, Phys. Rev. E., 69 (2004) 036126.
[2] O.M.P. Agrawal, A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vib. Control, (2007) 1269-1281.
[3] O.M.P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, J. Vib. Control, 14 (2008) 1291-1299.
[4] R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983) 201--210.
[5] R.L. Bagley and P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23 (1985) 918-925.
[6] D. Baleanu, O. Defterli, and O.M.P. Agrawal, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15 (2009) 547-597.
[7] G. Bohannan, Analog fractional order controller in temperature and motor control applications, J. Vibr. Control, 14 (2008) 1487-1498.
[8] K.B. Datta and B.M. Mohan, Orthogonal Functions in Systems and Control, World Scientific, Singapore, 1995.
[9] R.A. Devore and L.R. Scott, Error bounds for Gaussian quadrature and weighted-L1 polynomial approximation, SIAM J. Numer. Anal., 21 (1984) 400-412.
[10] K. Diethelm and N.J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004) 621--640.
[11] W. Grzesikiewicz, A. Wakulicz, and A. Zbiciak, Non-linear problems of fractional in modelling of mechanical systems, Int. J. Mech. Sci., 70 (2013) 90-89.
[12] J.H. He, Some applications of nonlinear fractional differential equations and their applications, Bull. Sci. Technol., 15 (1999) 86-90.
[13] M. Ichise,Y. Nagayanagi, and T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electroanal. Chem., 33 (1971) 253-265.
[14] H. Jafari, and H. Tajadodi, Fractional order optimal control problems via the operational matrices of Bernstein polynomials, U.P.B. Sci. Bull., Series A, 76(3) (2014) 115-128.
[15] Y. Jiang, X. Wang, and Y. Wang, On a stochastic heat equation with first order fractional noises and applications to finance, J. Math. Anal, Appl., 396 (2012) 656-669.
[16] E. Keshavarz, Y. Ordokhani, and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials, J. Vib. Control, 29 (2015) 1-15.
[17] R. Lewandowski, and B. Chorazyczewski, Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers, Comput. Struct., 88 (2010) 1-17.
[18] A. Lotfi, M. Dehghan, and S.A. Yousefi, A numerical technique for solving fractional optimal control problems, Comput. Math. Appl., 62 (2011) 1055-1067.
[19] A. Lotfi, S.A. Yousefi, and Mehdi Dehghanb, 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 (2013) 143-160
[20] R.L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004) 1-104.
[21] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fract. Fract. Calculus Contin. Mech., 387 (1997) 291--348.
[22] Y.A. Rossikhin and M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997) 15-67.
[23] N. Sebaa, Z.E.A. Fellah, W. Lauriks, and C. Depollier, Application of fractional calculus to ultrasonic wave propagation in human cancellous bone, Signal Process., 86 (2006) 2668-2677.
[24] V.E. Tarasov, Fractional vector calculus and fractional Maxwells equations, Annals of Physics, 323 (2008) 2756-2778.
[25] J.A. Tenreiro Machado, P. Stefanescu, O. Tintareanu, and D. Baleanu, Fractional calculus analysis of the cosmic microwave background, Rom. Rep. Phys., 65 (2013) 316-323.
[26] E. Tohidi and H. Saberi Nik, A Bessel collocation method for solving fractional optimal control problems, Appl. Math. Model., 39(2) (2015) 455-465.
[27] C. Tricaud and Y.Q. Chen, An approximation method for numerically solving fractional order optimal control problems of general form, Comput. Math. Appl., 59 (2010) 1644-1655.
[28] S.A. Yousefi, A. Lotfi, and M. Dehghan, The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems, J. Vib. Control, 13 (2011) 1-7.
[29] S.B. Yuste, L. Acedo, and K. Lindenberg, Reaction front in an A + B → C reaction-subdiffusion process, Phys. Rev. E., 69 (2004) 036126.
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