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Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

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Abstract

A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems (NOCPs). In fact, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s Maximum Principle (PMP), is solved by spectral homotopy analysis method (SHAM). For the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. It is indicated that Legendre collocation gives the same numerical results with Chebyshev collocation. Comparisons are made between SHAM, Matlab bvp4c generated results and results from literature such as homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM) and differential transformations.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    H. Saberi Nik & S. Effati

  2. Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP), Ferdowsi University of Mashhad, Mashhad, Iran

    H. Saberi Nik & S. Effati

  3. School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville, 3209, Pietermaritzburg, South Africa

    S. S. Motsa

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

    M. Shirazian

Authors
  1. H. Saberi Nik
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  2. S. Effati
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  3. S. S. Motsa
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  4. M. Shirazian
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Correspondence to H. Saberi Nik.

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Saberi Nik, H., Effati, S., Motsa, S. et al. Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems. Numer Algor 65, 171–194 (2014). https://doi.org/10.1007/s11075-013-9700-4

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  • DOI: https://doi.org/10.1007/s11075-013-9700-4

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