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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Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)
Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)
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)
Tang, X., Liu, Z., Wang, X.: Integral fractional pseudospectral methods for solving fractional optimal control problems. Automatica 62, 304–311 (2015)
Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055–1067 (2011)
Cresson, J.: Fractional Calculus in Analysis, Dynamics and Optimal Control. Nova Science Publishers, New York (2014)
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)
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)
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
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
Caputo, M.: Elasticitàe dissipazione. Zanichelli, Bologna (1969)
Jiao, Z., Chen, Y., Podlubny, I.: Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives. Springer, London (2012)
Hartley, T.T., Lorenzo, C.F.: Fractional system identification: an approach using continuous order-distributions. Technical report NASA (1999)
Eab, C.H., Lim, S.C.: Fractional Langevin equations of distributed order. Phys. Rev. E 83, 031136 (2011)
Lorenzo, C.F., Hartley, T.T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)
Atanackovic, T.M.: A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mech. 159, 77–86 (2002)
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)
Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Annali dellUniversita di Ferrara 41, 73–84 (1995)
Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001)
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
Chechkin, A., Gorenflo, R., Sokolov, I.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66, 046129 (2002)
Sokolov, I., Chechkin, A., Klafter, J.: Distributed-order fractional kinetics. Acta Phys. Pol. B 35, 1323–1341 (2004)
Meerschaert, M.M., Scheffler, H.P.: Stochastic model for ultraslow diffusion. Stoch. Process. Appl. 116, 1215–1235 (2006)
Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340, 252–281 (2008)
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)
Ford, N., Morgado, M.: Distributed order equations as boundary value problems. Comput. Math. Appl. 64, 2973–2981 (2012)
Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12, 409–422 (2009)
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)
Mainardi, F., Pagnini, G., Mura, A., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14, 1267–1290 (2008)
Meerschaert, M.M., Nane, E., Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379, 216–228 (2011)
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)
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)
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)
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)
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)
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
Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction–diffusion equations. J. Comput. Appl. Math. 275, 216–227 (2015)
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)
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)
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)
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)
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)
Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4, 531–542 (2001)
Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225, 96–104 (2009)
Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11–22 (2014)
Mashayekhi, S., Razzaghi, M.: Numerical solution of distributed order fractional differential equations by hybrid functions. J. Comput. Phys. 315, 169–181 (2016)
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)
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)
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)
Carnahan, B., Luther, H.A., Wilkes, J.O.: Applied Numerical Methods. Wiley, New York (1969)
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, London (1978)
Agrawal, O.P.: A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control 14, 1291–1299 (2008)
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)
Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A Math. Gen. 39, 10375–10384 (2006)
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)
Sage, A.P., White, C.C.: Optimal Systems Control. Prentice-Hall, Englewood Cliffs, NJ (1977)
Baleanu, D., Tenreiro Machado, J.A., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2012)
Alizadeh, A., Effati, S.: An iterative approach for solving fractional optimal control problems. J. Vib. Control (2016). https://doi.org/10.1177/1077546316633391
Zeid, S.S., Yousefi, M.: Approximated solutions of linear quadratic fractional optimal control problems. J. Appl. Math. 12, 83–94 (2016)
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
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)
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)
Bhrawy, A.H., Ezz-Eldien, S.S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo 53, 521–543 (2016)
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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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DOI: https://doi.org/10.1007/s11071-017-4038-4