Document Type : Research Articles
Authors
- Seyed Mehdi Shafiof ^{} ^{1}
- Javad Askari Marnani ^{2}
- Maryam Shamssolary ^{1}
^{1} Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
^{2} Faculty of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran
Abstract
This article aims to introduce a modern numerical method based on the hybrid functions, consisting of the Bernoulli polynomials and Block-Pulse functions. An indirect approach is proposed for solving the fractional optimal control problems (FOCPs). Firstly, the two-point boundary value problem (TPBVP) is calculated for a class of FOCPs, including integer-fractional derivatives, leading to a system of fractional differential equations (FDEs), which have the left and right-sided Caputo fractional derivatives (CFD). Therefore, a new approach is proposing to achieve the left Riemann-Liouville fractional integral (LRLFI) and right Riemann-Liouville fractional integral (RRLFI) operators for Bernoulli hybrid functions. Then, hybrid functions approximation, LRLFI, and RRLFI operators, and the collocation method are used to solve the TPBVP. The error bounds for the hybrid function and LRLFI and RRLFI operators are also presented. Moreover, the convergence of the proposed method is proved. Finally, the simplicity and accuracy of the method are illustrated using some numerical examples.
Keywords
- Bernoulli Hybrid functions
- Fractional optimal control
- Riemann-Liouville fractional Integral operators. Collocation method
Main Subjects
Physics, World Scientific Publishing, River Edge, NJ,
USA, 2000.
[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
Theory and Applications of Fractional Differential
Equations, Vol. 204, (North-Holland Mathematics
Studies), Elsevier Science Inc. New York, NY, 2006.
[3] I. Podlubny, Fractional Differential Equations: An
Introduction to Fractional Derivatives, Fractional
Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, 1999.
[4] S. G. Samko, A. A. Kilbas, and O. I. Marichev,
Fractional Integrals and Derivatives: Theory and
Applications, Switzerland: Gordon and Breach Science
Publishers, 1993.
[5] 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, Vol. 50, No. 1, pp. 15-67, 1997.
[6] R. L. Magin, “Fractional calculus in Bioengineering,”
Crit. Rev. Biomed. Eng, Vol. 32, No. 1, pp. 1-104, 2004.
[7] G. Jumarie, “Modelling fractional stochastic systems as
non-random fractional dynamics driven by Brownian
motions,” APPL. Math. Model, Vol. 32, No. 5, pp. 836-
859, 2008.
[8] B. Alkahtani, V. Gulati, and A. Kalman, “Application
of sumudu transform in generalized fractional reaction
diffusion equation,” Int. J. Appl. Comput. Math, Vol. 2,
pp. 387-394, 2016.
[9] A. S. Shaikh, I. N. Shaikh, and K. S. Nisar, “A
mathematical model of COVID-19 using fractional
derivative: outbreak in India with dynamics of
transmission and control,” Adv. Differ. Equ, Vol. 373,
2020.
[10] A. Atangana, “Modelling the spread of COVID-19 with
new fractal-fractional operators: Can the lockdown save
mankind before vaccination?,” Chaos Solitons
Fractals, Vol. 136, 2020.
[11] Z. Shabani and H. Tajadodi, “A numerical scheme for
constrained optimal control problems,” International
Journal of Industrial Electronics, Control and
Optimization, Vol. 2, No. 3, pp. 233-238, 2019.
[12] M. Alipour and P. Omidiniya, “A Semi-Analytic
Method for Solving a Class of Non-Linear Optimal
Control Problems,” International Journal of Industrial Electronics, Control and Optimization, Vol. 3, No. 4,
pp. 415-429, 2020.
[13] O. P. Agrawal, “A general formulation and solution
scheme for fractional optimal control problems,”
Nonlinear Dyn, Vol. 38, No. (1-4), pp. 323-337, 2004.
[14] O. P. Agrawal and D. Baleanu, “A Hamiltonian
Formulation and a Direct Numerical Scheme for
Fractional Optimal Control Problems,” Journal of
Vibration and Control, Vol. 13, No. (9-10), pp. 1269–
1281. 2007.
[15] O. P. Agrawal, “A quadratic numerical scheme for
fractional optimal control problem,” J. Dyn. Sys. Meas.
Control, Vol. 130, No. 1, pp. 011010-011010-6, 2008.
[16] S. A. Yusefi, A. Lotfi, and M. Dehghan, “The use of a
Legendre multiwavelet collocation method for solving
the fractional optimal control problem,” Journal of
Vibration and Control, Vol. 17, No. 13, pp. 2059-2065,
2011.
[17] A. Alizadeh, S. Effati, and A. Heidari, “Numerical
schemes for fractional optimal control problems,” J.
Dyn. Sys. Meas. Control, Vol. 139, No. 8, pp. 1-17,
2017.
[18] S. Ghasemi and A. Nazemi, “A fractional power series
neural network for solving a class of fractional optimal
control problems with equality and inequality
constraints,” Network: Computation in Neural Systems,
Vol. 30, No. (1-4), pp. 148-175, 2019.
[19] S. S. Zeid, S. Effati, and A. V. Kamyad,
“Approximation methods for solving fractional optimal
control problems,” Comp. Appl. Math, Vol. 37, pp. 158-
182, 2017.
[20] S. Soradi-Zeid, “Efficient radial basis functions
approaches for solving a class of fractional optimal
control problems,” Computational and Applied
Mathematics, Vol. 39, No. 1, 2020.
[21] H. R. Marzban and F. Malakoutikhah, “Solution of
delay fractional optimal control problems using a hybrid
of Block-Pulse functions and orthonormal Taylor
polynomials,” Journal of the Franklin Institute, vol.
356, No. 15, pp. 8182–8215, 2019.
[22] E. Keshavarz, Y. Ordokhani, and M. Razzaghi, “A
numerical solution for fractional optimal control
problems via Bernoulli polynomial,” Journal of
Vibration and Control, Vol. 22, No. 18, pp. 3889-3903,
2015.
[23] C. Phang, N. F. Ismail, A. Isah, and J. R. Loh, “A new
efficient numerical scheme for solving fractional
optimal control problems via a Genocchi operational
matrix of integration,” Journal of Vibration and
Control, Vol. 24, No. 14, pp. 3036-3048, 2017.
[24] H. Dehestani, Y. Ordokhani, and M. Razzaghi,
“Fractional-order Bessel wavelet functions for solving
variable order fractional optimal control problems with
estimation error,” Int J Syst Sci, Vol. 51, No. 6, pp.
1032–1052, 2020.
[25] K. Maleknejad and E. Saeedpoor, “An efficient method
based on hybrid functions for Fredholm integral
equation of the first kind with convergence analysis,”
Applied Mathematics and Computation, Vol. 304, pp.
93-102, 2017.
[26] S. Mashayekhi and M. Razzaghi, “An approximate
method for solving fractional optimal control problems
by hybrid functions,” Journal of Vibration and Control,
Vol. 24, No. 9, pp. 1621-1631, 2018.
“A hybrid functions numerical scheme for fractional
optimal control problem: Application to nonanalytic
dynamic systems,” Journal of Vibration and Control,
Vol. 24, No. 21, pp. 5030-5043, 2018.
[28] V. Taherpour, M. Nazari, and A. Nemati, “A new
numerical Bernoulli polynomial method for solving
fractional optimal control problems with vector
components,” Computational Methods for Differential
Equations, Vol. 9, No. 2, pp. 446-466, 2021.
[29] Y. Zhou, Basic Theory of Fractional Differential
Equations, World Scientific Publishing, Co. Pte. Ltd,
2014.
[30] C. Viola, An Introduction to Special Functions,
Springer International Publishing, Switzerland, 2016.
[31] E. Kreyszig, Introductory Functional Analysis with
Applications, Vol. 81, John Wiley & Sons, New York,
NY, USA, 1989.
[32] N. H. Sweilam and T. M. Alajmi, “Legendre spectral
collocation method for solving some type of fractional
optimal control problem,” J. Adv. Res, Vol. 6, No. 3, pp.
393-403, 2015.
[33] A. Lotfi and S. A. Yusefi, “Epsilon-Ritz method for
solving a class of fractional constrained optimization
problems,” J. Optim. Theory Appl, Vol. 163, No. 3, pp.
884-899, 2014.
[34] A. Alizadeh and S. Effati, “Modified Adomian
decomposition method for solving fractional optimal
control problems,” Transactions of the Institute of
Measurement and Control, Vol. 40, No. 6, pp. 2054–
2061, 2017.
[35] A. H. Bhrawy, S. S. Ezz-Eliden, E. H. Doha, M. A.
Abdelkawy, and D. Baleanu, “Solving fractional
optimal control problems with in Chebyshev-Legendre
operational technique,” Internat. J. Control, Vol. 90,
No. 6, pp. 1230-1244, 2017.
[36] K. Rabiei and K. Parand, “Collocation method to solve
inequality-constrained optimal control problems of
arbitrary order,” Engineering with Computers, Vol. 36,
pp. 115–125, 2020.
[37] H. Jaddu, “Direct solution of nonlinear optimal control
problems using quasilinearization and Chebyshev poly-
nomials,” Journal of the Franklin Institute, Vol. 339,
pp. 479–498, 2002.
[38] S. Sabermahani and Y. Ordokhani, “Fibonacci wavelets
and Galerkin method to investigate fractional optimal
control problems with bibliometric analysis,” Journal of
Vibration and Control, pp. 1-15, 2020.