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A Legendre collocation method for distributed-order fractional optimal control problems

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Abstract

In many dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also possess a continuous nature in a sense that their order is distributed over a given range. This paper is concerned with the optimization of systems whose governing equations contain a fractional derivative of distributed order, in the Caputo sense. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration-by-parts formula, the necessary optimality conditions are derived in terms of a nonlinear two-point distributed-order fractional boundary value problem. A Legendre spectral collocation method is developed for solving such problem. The solution method involves the use of three-term recurrence relations for both the left- and right-sided fractional integrals of the shifted Legendre polynomials. The convergence of the proposed method is discussed. The optimal profiles show the performance of the numerical solution and the effect of the fractional derivatives in the optimal results.

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References

  1. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  2. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Toledo-Hernandez, R., Rico-Ramirez, V., Rico-Martinez, R., Hernandez-Castro, S., Diwekar, U.M.: A fractional calculus approach to the dynamic optimization of biological reactive systems. Part II: numerical solution of fractional optimal control problems. Chem. Eng. Sci. 117, 239–247 (2014)

    Article  Google Scholar 

  4. Tang, X., Liu, Z., Wang, X.: Integral fractional pseudospectral methods for solving fractional optimal control problems. Automatica 62, 304–311 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055–1067 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cresson, J.: Fractional Calculus in Analysis, Dynamics and Optimal Control. Nova Science Publishers, New York (2014)

    Google Scholar 

  7. Singha, N., Nahak, C.: An efficient approximation technique for solving a class of fractional optimal control problems. J. Optim. Theory Appl. 174, 785–802 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zaky, M.A., Tenreiro Machado, J.A.: On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)

    Article  MathSciNet  Google Scholar 

  9. Baleanu, D., Jajarmi, A., Hajipour, M.: A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel. J. Optim. Theory Appl. (2017). https://doi.org/10.1007/s10957-017-1186-0

    MathSciNet  MATH  Google Scholar 

  10. Zeid, S.S., Effati, S., Kamyad, A.V.: Approximation methods for solving fractional optimal control problems. Comp. Appl. Math. (2017). https://doi.org/10.1007/s40314-017-0424-2

    MATH  Google Scholar 

  11. Caputo, M.: Elasticitàe dissipazione. Zanichelli, Bologna (1969)

    Google Scholar 

  12. Jiao, Z., Chen, Y., Podlubny, I.: Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives. Springer, London (2012)

    Book  MATH  Google Scholar 

  13. Hartley, T.T., Lorenzo, C.F.: Fractional system identification: an approach using continuous order-distributions. Technical report NASA (1999)

  14. Eab, C.H., Lim, S.C.: Fractional Langevin equations of distributed order. Phys. Rev. E 83, 031136 (2011)

    Article  MathSciNet  Google Scholar 

  15. Lorenzo, C.F., Hartley, T.T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Atanackovic, T.M.: A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mech. 159, 77–86 (2002)

    Article  MATH  Google Scholar 

  17. Atanackovic, T.M., Pilipovic, S., Zorica, D.: Time distributed-order diffusion-wave equation. I. Volterra-type equation. Pro. R. Soc. A Math. Phys. Eng. Sci. 465, 1869–1891 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Annali dellUniversita di Ferrara 41, 73–84 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Zaky, M.A.: A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comp. Appl. Math. (2017). https://doi.org/10.1007/s40314-017-0530-1

    Google Scholar 

  21. Chechkin, A., Gorenflo, R., Sokolov, I.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66, 046129 (2002)

    Article  Google Scholar 

  22. Sokolov, I., Chechkin, A., Klafter, J.: Distributed-order fractional kinetics. Acta Phys. Pol. B 35, 1323–1341 (2004)

    Google Scholar 

  23. Meerschaert, M.M., Scheffler, H.P.: Stochastic model for ultraslow diffusion. Stoch. Process. Appl. 116, 1215–1235 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340, 252–281 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sandev, T., Chechkin, A.V., Korabel, N., Kantz, H., Sokolov, I.M., Metzler, R.: Distributed-order diffusion equations and multifractality: models and solutions. Phys. Rev. E 92, 042117 (2015)

    Article  MathSciNet  Google Scholar 

  26. Ford, N., Morgado, M.: Distributed order equations as boundary value problems. Comput. Math. Appl. 64, 2973–2981 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12, 409–422 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations. Part I. Int. J. Appl. Math. 2(7), 865–882 (2000)

    MathSciNet  MATH  Google Scholar 

  29. Mainardi, F., Pagnini, G., Mura, A., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14, 1267–1290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Meerschaert, M.M., Nane, E., Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379, 216–228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Z., Luchko, Y., Yamamoto, M.: Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem. Comput. Math. Appl. 73, 1041–1052 (2017)

    Article  MathSciNet  Google Scholar 

  32. Fernández-Anaya, G., Nava-Antonio, G., Jamous-Galante, J., Muűoz-Vega, R., Hernéndez-Martínez, E.G.: Asymptotic stability of distributed order nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 48, 541–549 (2017)

    Article  MathSciNet  Google Scholar 

  33. Naranjani, Y., Sardahi, Y., Chen, Y., Sun, J.: Multi-objective optimization of distributed-order fractional damping. Commun. Nonlinear Sci. Numer. Simul. 24, 159–168 (2015)

    Article  MathSciNet  Google Scholar 

  34. Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algor. 75, 173–211 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ye, H., Liu, F., Anh, V.: Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys. 298, 652–660 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gao, G., Alikhanov, A.A., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. (2017). https://doi.org/10.1007/s10915-017-0407-x

    MathSciNet  MATH  Google Scholar 

  37. Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction–diffusion equations. J. Comput. Appl. Math. 275, 216–227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ford, N.J., Morgado, M.L., Rebelo, M.: An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time. Electron. Trans. Numer. Anal. 44, 289–305 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Ford, N.J., Morgado, M.L., Rebelo, M.: A numerical method for the distributed order time-fractional diffusion equation. In: Proceedings of the International Conference on Fractional Differentiation and Its Applications, pp. 1–6. IEEE (2014)

  40. Pimenov, V.G., Hendy, A.S., De Staelen, R.H.: On a class of non-linear delay distributed order fractional diffusion equations. J. Comput. Appl. Math. 318, 433–443 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Bu, W., Xiao, A., Zeng, W.: Finite difference/finite element methods for distributed-order time fractional diffusion equations. J. Sci. Comput. 72, 422–441 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Fan, W., Liu, F.: A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl. Math. Lett. 77, 114–121 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4, 531–542 (2001)

    MathSciNet  MATH  Google Scholar 

  44. Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225, 96–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11–22 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Mashayekhi, S., Razzaghi, M.: Numerical solution of distributed order fractional differential equations by hybrid functions. J. Comput. Phys. 315, 169–181 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: Petrov–Galerkin and spectral collocation methods for distributed order differential equations. SIAM J. Sci. Comput. 39(3), A1003–A1037 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lischke, A., Zayernouri, M., Karniadakis, G.E.: A Petrov–Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. SIAM J. Sci. Comput. 39(3), A922–A946 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Morgado, M., Rebelo, M., Ferrás, L., Ford, N.: Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method. Appl. Numer. Math. 114, 108–123 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  50. Carnahan, B., Luther, H.A., Wilkes, J.O.: Applied Numerical Methods. Wiley, New York (1969)

    MATH  Google Scholar 

  51. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, London (1978)

    MATH  Google Scholar 

  52. Agrawal, O.P.: A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control 14, 1291–1299 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Frederico, G.S., Torres, D.F.: Fractional optimal control in the sense of caputo and the fractional Noether’s theorem. Int. Math. Forum 3, 479–493 (2008)

    MathSciNet  MATH  Google Scholar 

  54. Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A Math. Gen. 39, 10375–10384 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  55. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods-Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  56. Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)

    MATH  Google Scholar 

  57. Sage, A.P., White, C.C.: Optimal Systems Control. Prentice-Hall, Englewood Cliffs, NJ (1977)

    MATH  Google Scholar 

  58. Baleanu, D., Tenreiro Machado, J.A., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)

    Book  MATH  Google Scholar 

  59. Alizadeh, A., Effati, S.: An iterative approach for solving fractional optimal control problems. J. Vib. Control (2016). https://doi.org/10.1177/1077546316633391

    Google Scholar 

  60. Zeid, S.S., Yousefi, M.: Approximated solutions of linear quadratic fractional optimal control problems. J. Appl. Math. 12, 83–94 (2016)

    MathSciNet  Google Scholar 

  61. Sahu, P.K., Ray, S.S.: Comparison on wavelets techniques for solving fractional optimal control problems. J. Vib. Control (2016). https://doi.org/10.1177/1077546316659611

    Google Scholar 

  62. Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13, 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  63. Doha, E.H., Bhrawy, A.H., Baleanu, D., Ezz-Eldien, S.S., Hafez, R.M.: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 2015, 15 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  64. Bhrawy, A.H., Ezz-Eldien, S.S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo 53, 521–543 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to express his gratitude to the anonymous reviewers for their constructive comments, which shaped the paper into its final form.

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  1. Department of Applied Mathematics, National Research Centre, Dokki, Giza, 12622, Egypt

    Mahmoud A. Zaky

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Correspondence to Mahmoud A. Zaky.

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Zaky, M.A. A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn 91, 2667–2681 (2018). https://doi.org/10.1007/s11071-017-4038-4

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