In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article

Authors

Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran‎.

10.30473/coam.2023.68826.1243

Abstract

This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and Jacobi polynomials. ‎First‎, ‎the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated‎. ‎ Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions‎. ‎ Next, ‎using the dynamical system and applying the fractional integral operator‎, ‎an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives‎. ‎ Subsequently‎, ‎all the given approximations are substituted into the performance index‎. ‎Finally‎, ‎the optimality conditions transform the problem into a system of algebraic equations‎. ‎An error upper bound of the approximation of a function based on the fractional hybrid functions is provided‎. ‎The method is applied to several numerical examples‎, and ‎the experimental results confirm the efficiency and capability of the method.  Furthermore, they demonstrate a good agreement between the approximate and exact solutions‎. ‎

Keywords

Main Subjects

[1] Agrawal, O.P. (2004). “A general formulation and solution scheme for fractional optimal control problems”, Nonlinear Dynamics, 38, 323-337.
[2] Agrawal, O.P. (2008). “A formulation and numerical scheme for fractional optimal control problems”, Journal of Vibration and Control, 14(9-10), 1291-1299.
[3] Agrawal, O.P. (2008). “A quadratic numerical scheme for fractional optimal control problems”, ASME Journal of Dynamic Systems, Measurement and Control, 130(1), 011010.
[4] Alizadeh, A., Effati, S. (2018). “An iterative approach for solving fractional optimal control problems”, Journal of Vibration and Control, 24(1), 18-36.
[5] Barikbin, Z., Keshavarz, E. (2020). “Solving fractional optimal control problems by new Bernoulli wavelets operational matrices”, Optimal Control Applications and Methods, 41(4), 1188-1210.
[6] Behroozifar, M., Habibi, N.(2018).“ A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials”, Journal of Vibration and Control,
[7] Bhrawy, A.H., Ezz-Eldien, S.S., Doha, E.H., Abdelkawy, M.A., Baleanu, D. (2017). “Solving fractional optimal control problems within a Chebyshev–Legendre operational technique”, International Journal of Control, 90(6), 1230-1244.
[8] DeVore, R.A., Scott, L.R. (1984). “Error bounds for Gaussian quadrature and weighted-L1 polynomial approximation”, SIAM Journal on Numerical Analysis, 21(2), 400-412.
[9] Ezz-Eldien, S.S., Doha, E.H., Baleanu, D., Bhrawy, A.H. (2017). “A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems”, Journal of Vibration and Control, 23(1), 16-30.
[10] Heydari, M.H., Avazzadeh, Z., Cattani, C. (2020). “Taylor’s series expansion method for nonlinear variable-order fractional 2D optimal control problems”, Alexandria Engineering Journal, 59(6), 4737-4743.
[11] Li, C., Cai, M. (2019). “Theory and numerical approximations of fractional integrals and derivatives”, Society for Industrial and Applied Mathematics.
[12] Lotfi, A., Yousefi, S.A., Dehghan, M. (2013). “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”, Journal of Computational and Applied Mathematics, 250, 143-160.
[13] Marzban, H.R., Razzaghi, M.(2006).“ Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series”, Journal of Sound and Vibration, 292(3-5), 954-963.
[14] Mashayekhi, S., Razzaghi, M. (2015). “Numerical solution of nonlinear fractional integro differential equations by hybrid functions”, Engineering Analysis with Boundary Elements, 56, 81-89.
[15] Mashayekhi, S., Razzaghi, M. (2016). “Numerical solution of distributed order fractional differential equations by hybrid functions”, Journal of Computational Physics, 315, 169-181.
[16] Mohammadi, F., Moradi, L., Baleanu, D., Jajarmi, A. (2018). “A hybrid functions numerical scheme for fractional optimal control problems: Application to nonanalytic dynamic systems”, Journal of Vibration and Control, 24(21), 5030-5043.
[17] Naidu, D.S. (2003). “Optimal control systems, Electrical engineering textbook series”, CRC Press.
[18] Nemati, S. (2016). “A spectral method based on the second kind Chebyshev polynomials for solving a class of fractional optimal control problems”‎, ‎Sahand Communications in Mathematical Analysis, 4(1), 15-27.
[19] Nemati, S., Lima, P.M., Torres, D.F. (2019). “A numerical approach for solving fractional optimal control problems using modified hat functions”, Communications in Nonlinear Science and Numerical Simulation, 78, 104849.
[20] Podlubny, I. (1999). “Fractional differential equations”. Mathematics in Science and Engineering, Academic press, New York.
[21] Postavaru, O., Toma, A. (2021). “Numerical solution of two-dimensional fractional-order partial differential equations using hybrid functions”, ‎Partial Differential Equations in Applied Mathematics‎, ‎4‎, ‎100099‎.
[22] Postavaru, O., Toma, A. (2022). “A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems”, Mathematics and Computers in Simulation, 194, 269-284.
[23] Rabiei, K., Razzaghi, M. (2023). “An approach to solve fractional optimal control problems via fractional-order Boubaker wavelets”, Journal of Vibration and Control, 29(7-8), 1806-1819.
[24] Razzaghi, M., Marzban, H.R. (2000). “A hybrid analysis direct method in the calculus of variations”, International Journal of Computer Mathematics, 75(3), 259-269.
[25] Razzaghi, M., Marzban, H.R. (2000). “Direct method for variational problems via hybrid of block-pulse and Chebyshev functions”, Mathematical Problems in Engineering, 6, 85-97.
[26] Shen, J., Tang, T., Wang, L.L., Shen, J., Tang, T., Wang, L.L. (2011). “Orthogonal polynomials and related approximation results”, Spectral Methods: Algorithms, Analysis and Applications, 47-140.
[27] Tricaud, C., Chen, Y. (2010). “An approximate method for numerically solving fractional order optimal control problems of general form”, Computers & Mathematics with Applications, 59(5), 1644-1655.
[28] Xiaobing, P., Yang, X., Skandari, M.H.N., Tohidi, E., Shateyi, S. (2022). “A new high accurate approximate approach to solve optimal control problems of fractional order via efficient basis functions”, Alexandria Engineering Journal, 61(8), 5805-5818.
[29] Yousefi, S.A., Lotfi, A., Dehghan, M. (2011). “The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems”, Journal of Vibration and Control, 17(13), 2059-2065.
[30] Zhang, J., Tang, Y., Liu, F., Jin, Z., Lu, Y. (2021). “Solving fractional differential equation using block pulse functions and Bernstein polynomials”, Mathematical Methods in the Applied Sciences, 44(7), 5501-5519.
[31] Zhang, B., Tang, Y., Zhang, X. (2021). “Numerical solution of fractional differential equations using hybrid Bernoulli polynomials and block pulse functions”, Mathematical Sciences, 15, 293-304.