Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations

  • Kobra Rabiei
  • Yadollah Ordokhani
  • Esmaeil Babolian


In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.


Fractional-order Legendre functions Optimal control problem Fractional integro-differential equations Operational matrix Convergence analysis 



We would like to thank the referees for their helpful suggestions to improve the earlier version of this article.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Kobra Rabiei
    • 1
  • Yadollah Ordokhani
    • 1
  • Esmaeil Babolian
    • 2
  1. 1.Department of Mathematics, Faculty of Mathematical SciencesAlzahra UniversityTehranIran
  2. 2.Department of Computer Science, Faculty of Mathematical Sciences and ComputerKharazmi UniversityTehranIran

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