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Generalized Bernoulli Polynomials: Solving Nonlinear 2D Fractional Optimal Control Problems

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Abstract

This work develops an optimization method based on a new class of basis function, namely the generalized Bernoulli polynomials (GBP), to solve a class of nonlinear 2-dim fractional optimal control problems. The problem is generated by nonlinear fractional dynamical systems with fractional derivative in the Caputo type and the Goursat–Darboux conditions. First, we use the GBP to approximate the state and control variables with unknown coefficients and parameters. Afterwards, we substitute the obtained values for the variables and parameters in the objective function, nonlinear fractional dynamical system and Goursat–Darboux conditions. The 2-dim Gauss–Legendre quadrature rule together with a fractional operational matrix construct a constrained problem, that is solved by the Lagrange multipliers method. The convergence of the GBP method is proved and its efficiency is demonstrated by several examples.

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Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    H. Hassani

  2. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 431, 4249-015, Porto, Portugal

    J. A. Tenreiro Machado

  3. Laboratory for Intelligent Computing and Financial Technology, Department of Mathematical Science, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, Jiangsu, China

    Z. Avazzadeh

  4. Department of Mathematics, Yasouj University, Yasouj, Iran

    E. Naraghirad

  5. Faculty of Mathematics, Shahrekord University, Shahrekord, Iran

    M. Sh. Dahaghin

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  1. H. Hassani
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  2. J. A. Tenreiro Machado
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  3. Z. Avazzadeh
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  4. E. Naraghirad
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  5. M. Sh. Dahaghin
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Hassani, H., Machado, J.A.T., Avazzadeh, Z. et al. Generalized Bernoulli Polynomials: Solving Nonlinear 2D Fractional Optimal Control Problems. J Sci Comput 83, 30 (2020). https://doi.org/10.1007/s10915-020-01213-0

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