Abstract
We give the hybrid of block-pulse function and generalized Mott polynomials (HBGMP) and use it to solve delay fractional optimal control problems (DFOCPs). First, we develop a method for computing the exact formula of the Riemann–Liouville fractional integral operator of the HBGMP by using the regularized beta function. Next, the DFOCPs will be transformed into parameter optimization problems. By imposing the optimality conditions, we reduce the problem to algebraic equations. Several examples are given to show the advantages of the method.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Ding, Y., Wang, Z., Ye, H.: Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Cont. Syst. Technol. 20(3), 763–769 (2011)
Benson, D.A., Meerschaert, M.M., Revielle, J.: Fractional calculus in hydrologicmodeling: a numerical perspective. Adv. Water. Resour. 51, 479–497 (2013)
Larsson, S., Racheva, M., Saedpanah, F.: Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity. Comput. Methods. Appl. Mech. Eng. 283, 196–209 (2015)
Popovic, J.K., Spasic, D.T., Tosic, J., Kolarovic, J.L., Malti, R., Mitic, I.M., Pilipovic, S., Atanackovic, T.M.: Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukaemia. Commun. Nonlinear. Sci. Numer. Simul. 22, 451–471 (2015)
Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73, 5–59 (1996)
Sierociuk, D., Dzielinski, A., Sarwas, G., Petras, I., Podlubny, I., Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculusPhil. Trans. R. Soc. A. 371, 20120146 (2013). https://doi.org/10.1098/rsta.2012.0146
Sun, L., Chen, L.: Free vibrations of a taut cable with a general viscoelastic damper modeled by fractional derivatives. J. Sound. Vib. 335, 19–33 (2015)
Mainardi, F.: Fractional Calculus some basic problems in continuum and statistical mechanics Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag: New York. 291-348 (1997)
Bohannan, G.: Analog fractional order controller in temperature and motor control applications. J. Vib. Contr. 14, 1487–1498 (2008)
Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1–104 (2004)
Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F., Cattani, C.: Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun. Nonlinear Sci. Numer. Simul. 19(1), 37–48 (2014)
Bryson, A.E., Ho, Y.C.: Applied Optimal Control Optimization, Estimation, and Control. Blaisdell Publishing Company, Waltham (1975)
Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966)
Rabiei, K., Ordokhani, Y., Babolian, E.: The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear. Dyn. 88(2), 1013–1026 (2017)
Nemati, S., Lima, P.M., Torres, D.F.M.: A numerical approach for solving fractional optimal control problems using modified hat functions. Commun. Nonlinear Sci. Numer. Simul. 78, 104849 (2019)
Driver, R.D.: Ordinary and Delay Differential Equations. Applied Mathematical Sciences. Springer, New York (1977)
Kuang, E.: Delay Differential Equations with Applications in Population Dynamics. Acadamic Press, Boston (1993)
Sweilam, N.H., AL-Mekhlafi, S.M.: Optimal control for a time delay multi-strain tuberculosis fractional model: a numerical approach. IMA J. Math. Control Inf 1(36), 317–340 (2019). https://doi.org/10.1093/imamci/dnx046
Das, P., Das, S., Das, P., Rihan, F.A., Uzuntarla, M., Ghosh, D.: Optimal control strategy for cancer remission using combinatorial therapy: a mathematical model-based approach. Chaos Solit. Fract. 145, 109 (2021). https://doi.org/10.1016/j.chaos.2021.110789
Sabermahani, S., Ordokhani, Y., Yousefi, S.A.: Fractional-order Lagrange polynomials: an application for solving delay fractional optimal control problems. Trans. Inst. Meas. 41(11), 2997–3009 (2019). https://doi.org/10.1177/0142331218819048
Singh, V.K., Pandey, R.K., Singh, S.: A stable algorithm for Hankel transforms using hybrid of block-pulse and Legendre polynomials. Comput. Phys. Commun. 181, 1–10 (2010)
Razzaghi, M., Marzban, H.R.: Direct method for variational problems via hybrid of block-pulse and Chebyshev functions. Math. Prob. Eng. 6, 85–97 (2000)
Marzban, H.R., Razzaghi, M.: Analysis of time-delay systems via hybrid of block-pulse functions and Taylor series. J. Vib. Contr. 11, 1455–1468 (2005)
Mashayekhi, S., Ordokhani, Y., Razzaghi, M.: Hybrid functions approach for nonlinear constrained optimal control problems. Commun. Nonlin. Sci. Numer. Simulat. 17, 1831–1843 (2012)
Mashayekhi, S., Razzaghi, M.: Numerical solution of distributed order fractional differential equations by hybrid functions. J. Comput. Phys. 315, 169–181 (2016)
Toan, P.T., Vo, T.N., Razzaghi, M.: Taylor wavelet method for fractional delay differential equations. Eng. Comput. 9, 1–10 (2019)
Vichitkunakorn, P., Vo, T.N., Razzaghi, M.: A numerical method for fractional pantograph differential equations based on Taylor wavelets. Trans. Inst. Meas. 42, 1334–1344 (2010)
Mott, N.F.: the polarisation of electrons by double scattering. In: proceedings of the royal society of London. Series A Containing Papers of a Mathematical and Physical Character. 135(827): 429-458 (1932)
Razzaghi, M., Elnagar, G.: Linear quadratic optimal control problems via shifted Legendre state parametrization. Int. J. Syst. Sci. 25, 393–399 (1994)
Mashayekhi, S., Ordokhani, Y., Razzaghi, M.: Hybrid functions approach for optimal control of systems described by integro-differential equations. Appl. Math. Model. 37(5), 3355–3368 (2013)
Marzban, H.R., Malakoutikhah, F.: Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials. J. Franklin. Inst. 356(15), 8182–8215 (2019)
Li, C., Qian, D., Chen, Y.: On Riemann-Liouville and Caputo derivatives. Discrete. Dyn. Nat. Soc. 20, 11 (2011). https://doi.org/10.1155/2011/562494
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Kruchinin, D.V.: Explicit formula for generalized Mott polynomials. Adv. Stud. Contemp. Math. 24, 327–322 (2014)
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: with Formulas. graphs, and mathematical tables. Dover (1973)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: An efficient approximate method for solving delay fractional optimal control problems. Nonlinear. Dyn. 86(3), 1649–1661 (2016)
Rabiei, K., Ordokhani, Y., Babolian, E.: Boubaker functions and their applications in solving delay fractional optimal control problems. J. Vib. Contr. 24(15), 3370–3383 (2017)
Malmir, I.: A general framework for optimal control of fractional nonlinear delay systems by wavelets. Stat. Optim. Inf. Comput. 8(4), 858–875 (2020)
Jajarmi, A., Baleanu, D.: Suboptimal control of fractional-order dynamic systems with delay argument. J. Vib. Contr. 24(12), 2430–2446 (2018)
Lewis, Frank L., Vrabie, Draguna L., Syrmos, Vassilis L.: Optimal Control, 3rd edn., pp. 97–114. Wiley, New York (2012)
Bass, R.F.: Real Analysis for Graduate Students. Createspace Ind, Pub (2013)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods-Fundamentals in Single Domains. Springer, Berlin (2006)
Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)
Dadkhah, M., Farahi, M.H.: Optimal control of time delay systems via hybrid of block-pulse functions and orthonormal Taylor series. Int. J. Appl. Comput. Math. 2(1), 137–152 (2015)
Moradi, L., Mohammadi, F., Baleanu, D.: A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. J. Vib. Contr. 25(2), 310–324 (2019)
Banks, H.T., Burns, J.A.: Hereitary control problems: numerical methods based on averaging approximations. SIAM J. Control. Optim. 16(2), 169–208 (1978)
Hosseinpour, S., Nazemi, A., Tohidi, E.: Muntz-Legendre spectral collocation method for solving delay fractional optimal control problems. J. Comput. Appl. Math. 351, 344–363 (2019)
Safaie, E., Farahi, M.H., Farmani Ardehaie, M.: An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Comput. Appl. Math. 34(3), 831–846 (2014)
Haddadi, N., Ordokhani, Y., Razzaghi, M.: Optimal control of delay systems by using an hybrid functions approximation. J. Optim. Theory Appl. 153, 338–356 (2012)
Mohammadzadeh, R., Lakestani, M.: Optimal control of linear time-delay systems by a hybrid of block-pulse functions and biorthogonal cubic Hermite spline multi-wavelets. Optim. Control Appl. Methods. 39, 357–376 (2018)
Khellat, F.: Optimal control of linear time-delayed systems by linear Legendre multi-wavelets. J. Optim. Theory. Appl. 143(1), 107–121 (2009)
Safaie, E., Farahi, M.H.: An approximation method for numerical solution of multi-dimensional feedback delay fractional optimal control problems by Bernstein polynomials. IJNAO. 4(1), 77–94 (2014)
Malmir, I.: Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models. Appl. Math. Model. 69, 621–647 (2019)
Bhrawy, A.H., Ezz-Eldien, S.S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo. 53(4), 521–543 (2016)
Marzban, H.R., Razzaghi, M.: Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials. J. Frankl. Inst. 341, 279–293 (2004)
Nazemi, A., Shabani, M.M.: Numerical solution of the time-delayed optimal control problems with hybrid functions. IMA J. Math. Control. Inf. 32(3), 623–638 (2015)
Rakhshan, S.A., Effati, S.: Fractional optimal control problems with time-varying delay: a new delay fractional Euler-Lagrange equations. J. Franklin. Inst. 357, 5954–5988 (2020)
Acknowledgements
The authors wish to express their sincere thanks to the anonymous referees for valuable suggestions that improved the final version of the manuscript.
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rabiei, K., Razzaghi, M. Hybrid of block-pulse functions and generalized Mott polynomials and their applications in solving delay fractional optimal control problems. Nonlinear Dyn 111, 6469–6486 (2023). https://doi.org/10.1007/s11071-022-08177-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-08177-w