Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 536–555 | Cite as

Numerical Solution of Two-Dimensional Variable-Order Fractional Optimal Control Problem by Generalized Polynomial Basis

  • Fakhrodin Mohammadi
  • Hossein HassaniEmail author


This paper deals with an efficient numerical method for solving two-dimensional variable-order fractional optimal control problem. The dynamic constraint of two-dimensional variable-order fractional optimal control problem is given by the classical partial differential equations such as convection–diffusion, diffusion-wave and Burgers’ equations. The presented numerical approach is essentially based on a new class of basis functions with control parameters, called generalized polynomials, and the Lagrange multipliers method. First, generalized polynomials are introduced and an explicit formulation for their variable-order fractional operational matrix is obtained. Then, the state and control functions are expanded in terms of generalized polynomials with unknown coefficients and control parameters. By using the residual function and its 2-norm, the under consideration problem is transformed into an optimization one. Finally, the necessary conditions of optimality results in a system of algebraic equations with unknown coefficients and control parameters can be simply solved. Some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method.


Two-dimensional variable-order fractional optimal control problem Generalized polynomials Operational matrix Lagrange multipliers Optimization method 

Mathematics Subject Classification

34A08 49J20 41A58 49J21 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of HormozganBandar AbbasIran
  2. 2.Shahrekord UniversityShahrekordIran

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