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Collocation method to solve inequality-constrained optimal control problems of arbitrary order

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Abstract

In this paper, the generalized fractional order of the Chebyshev functions (GFCFs) based on the classical Chebyshev polynomials of the first kind is used to obtain the solution of optimal control problems governed by inequality constraints. For this purpose positive slack functions are added to inequality conditions and then the operational matrix for the fractional derivative in the Caputo sense, reduces the problems to those of solving a system of algebraic equations. It is shown that the solutions converge as the number of approximating terms increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach one. The applicability and validity of the method are shown by numerical results of some examples, moreover a comparison with the existing results shows the preference of this method.

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References

  1. Benson DA, Meerschaert MM, Revielle J (2013) Fractional calculus in hydrologic modeling: a numerical perspective. Adv. Water Resour 51:479–497

    Article  Google Scholar 

  2. Larsson S, Racheva M, Saedpanah F (2015) Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity. Comput. Method. Appl. Mech. Eng. 283:196–209

    Article  MathSciNet  Google Scholar 

  3. Gonzalez-Parra G, Arenas AJ, Chen-Charpentier BM (2014) A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1). Math. Methods Appl. Sci. 37(15):2218–2226

    Article  MathSciNet  Google Scholar 

  4. Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J. Vib. Control. 14:1487–1498

    Article  MathSciNet  Google Scholar 

  5. Yang XJ, Machado JAT (2015) A new insight into complexity from the local fractional calculus view point: modelling growths of populations, Math. Methods Appl. Sci. https://doi.org/10.1002/mma.3765

    Article  MathSciNet  Google Scholar 

  6. Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50:15–67

    Article  Google Scholar 

  7. Magin RL (2004) Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32:1–104

    Article  Google Scholar 

  8. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differential Equations. In: North-Holland Mathematics Studies, 204, Elsevier Science B.V, Amsterdam

  9. Pu YF, Siarry P, Zhou L, Zhang N (2014) A fractional partial differential equation based multiscale denoising model for texture image. Math. Methods Appl. Sci. 37(12):1784–1806

    Article  MathSciNet  Google Scholar 

  10. Jesus IS, Machado JAT (2008) Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3):263–282

    Article  MathSciNet  Google Scholar 

  11. Suarez IJ, Vinagre BM, Chen YQ (2008) A fractional adaptation scheme for lateral control of an AGV. J. Vib. Control 14:1499–1511

    Article  MathSciNet  Google Scholar 

  12. Agrawal OP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38:323–337

    Article  MathSciNet  Google Scholar 

  13. Agrawal OP (2007) A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problem. J. Vib. Control. 13:1269–1281

    Article  MathSciNet  Google Scholar 

  14. Lotfi A, Yousefi SA, 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. J. Comput. Appl. Math. 250:143–160

    Article  MathSciNet  Google Scholar 

  15. Zeid SS, Effati S, Kamyad AV (2016) Approximation methods for solving fractional optimal control problems, Computational and Applied Mathematics https://doi.org/10.1007/s40314-017-0424-2

    Article  MathSciNet  Google Scholar 

  16. Rabiei K, Ordokhani Y, Babolian E (2017) Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems, J. Vib. Control. https://doi.org/10.1177/1077546317705041

    Article  MathSciNet  Google Scholar 

  17. Almeida R, Torres DFM (2015) A discrete method to solve fractional optimal control problems, Nonlinear Dyn. 2015; 80(2), 1811-1816

    Article  MathSciNet  Google Scholar 

  18. Safaie E, Farahi M.H., Ardehaie M Farmani (2015) An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials, Computational and Applied Mathematics, 34(3) , 831-846

    Article  MathSciNet  Google Scholar 

  19. Rabiei K, Ordokhani Y, Babolian E (2017) The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dyn. 88(2):1013–1026

    Article  MathSciNet  Google Scholar 

  20. Drefus SF (1962) Variational problems with state variable inequality constraint. J. Math. Anal. Appl. 4:291–301

    Google Scholar 

  21. Mehra RK, Davis RE (1972) A generalized gradiant method for optimal control problems with inequality constraint and singular arcs. IEEE Transactions on Automatic control 17:69–72

    Article  Google Scholar 

  22. Khalid A, Huey J, Singhose W, Lawrence J, Frakes D (2006) Human operator performance testing using an input\(-\)shaped bridge crane. J. Dyn. Sys. Meas. Control. 128(4):835–841

    Article  Google Scholar 

  23. Tuan LA, Lee SG (2013) Sliding mode controls of double-pendulum crane systems. Journal of Mechanical Science and Technology 27(6):1863–1873

    Article  Google Scholar 

  24. Gong Q, Kang W, Ross IM (2006) A pseudospectral method for the optimal control of constrained feedback linearizable systems. IEEE Transactions on Automatic Control 51(7):1115–1129

    Article  MathSciNet  Google Scholar 

  25. Li M, Peng H (2016) Solutions of nonlinear constrained optimal control problems using quasilinearization and variational pseudospectral methods. ISA Transactions 62:177–192

    Article  Google Scholar 

  26. Wang X, Peng H, Zhang S, Chen B, Zhong W (2017) A symplectic pseudospectra method for nonlinear optimal control problems with inequality constraints, ISA Transactions https://doi.org/10.1016/j.isatra.2017.02.018

    Article  Google Scholar 

  27. Lu L, Liu Z, Jiang W, Luo J (2016) Optimal controls for fractional differential evolution hemivariational inequalities. Math. Methods Appl. Sci. 39(18):5452–5464

    Article  MathSciNet  Google Scholar 

  28. Alipour M, Rostamy D, Baleanu D (2013) Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vib. Control. 19:2523–2540

    Article  MathSciNet  Google Scholar 

  29. Rabiei K, Ordokhani Y (2018) Boubaker hybrid functions and their application to solve fractional optimal control and fractional variational problems. Application of Mathematics. 63(5):541–567

    Article  MathSciNet  Google Scholar 

  30. Bhrawy AH, Alofi AS (2013) The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Letters. 26:25–31

    Article  MathSciNet  Google Scholar 

  31. Parand K, Abbasbandy S, Kazem S, Rezaei AR (2011) An improved numerical method for a class of astrophysics problems based on radial basis functions, Phys. Scr. 83 (11)

    Article  Google Scholar 

  32. Doha EH, Bhrawy AH, Ezz-Eldien SS (2011) A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62:2364–2373

    Article  MathSciNet  Google Scholar 

  33. Parand K, Delkhosh M (2016) Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions. Ricerche Mat. 65(1):307–328

    Article  MathSciNet  Google Scholar 

  34. Darani MA, Nasiri M (2013) A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations. Comp. Meth. Differ. Equ. 1:96–107

    MATH  Google Scholar 

  35. Lancaster P (1969) Theory of Matrices. Academic Press, New York

    MATH  Google Scholar 

  36. Jaddu H (2002) Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials. J Frank Inst 339:479–498

    Article  MathSciNet  Google Scholar 

  37. Mashayekhi S, Ordokhani Y, Razzaghi M (2012) Hybrid functions approach for nonlinear constrained optimal control problems. Commun Nonlinear Sci Numer Simulat 17:1831–1843

    Article  MathSciNet  Google Scholar 

  38. Elnagar G, Kazemi MA, Razzaghi M (1995) The Pseudospectral Legendre method for discretizing optimal control problems. IEEE Transactions On Automatic Control. 40(10):1793–1796

    Article  MathSciNet  Google Scholar 

  39. Maleki M, Tirani M Dadkhah (2011) Chebyshev finite difference method for solving constrained quadratic optimal control problems. Journal of Mathematical Extension 52(1):1–21

    MathSciNet  MATH  Google Scholar 

  40. Vlassenbroeck J (1988) A Chebyshev polynomial method for optimal control with state constraints. Automatica. 24:499–506

    Article  MathSciNet  Google Scholar 

  41. Ordokhani Y, Razzaghi M (2005) Linear quadratic optimal control problems with inequality constraints via Rationalizes Haar functions, DCDIS Series B: Applications and Algorithms, 761-773

  42. Mashayekhi S, Razzaghi M (2018) An approximate method for solving fractional optimal control problems by hybrid functions. J. Vib. Control. 24(9):1621–1631

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We have to express our appreciation to the reviewers for their helpful comments which improve the quality of this work.

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Authors and Affiliations

  1. Department of Computer Sciences, Shahid Beheshti University, G.C, Tehran, Iran

    K. Rabiei & K. Parand

  2. Department of Cognitive Modelling, Cognitive and Brain Sciences, Shahid Beheshti University, G.C, Tehran, Iran

    K. Parand

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  1. K. Rabiei
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  2. K. Parand
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Correspondence to K. Rabiei.

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Rabiei, K., Parand, K. Collocation method to solve inequality-constrained optimal control problems of arbitrary order. Engineering with Computers 36, 115–125 (2020). https://doi.org/10.1007/s00366-018-0688-1

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  • DOI: https://doi.org/10.1007/s00366-018-0688-1

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