Soft Computing

, Volume 22, Issue 17, pp 5889–5899 | Cite as

A modified teaching–learning-based optimization for optimal control of Volterra integral systems

  • R. KhanduziEmail author
  • A. Ebrahimzadeh
  • M. Reza Peyghami
Methodologies and Application


This study aimed to utilize a novel modified approach based on teaching–learning-based optimization (MTLBO), to achieve an approximate solution of optimal control problem governed by nonlinear Volterra integro-differential systems. The scheme was based upon Chebyshev wavelet and its derivative operational matrix, which eventually led to a nonlinear programming problem (NLP). The resulted NLP was solved by the MTLBO. The novel algorithm used a heuristic mechanism to intensify learning on the best students in learner phase. The new strategy was applied to improve learners’ knowledge and to structure the MTLBO. The applicability and efficiency of the MTLBO were shown for three numerical examples. The proposed algorithm was compared with the traditional TLBO algorithm and the Legendre wavelets and collocation method in the literature. The experimental results showed that the proposed MTLBO not only obtained the high-quality solutions with respect to the absolute errors but also provided results with the high speed of convergence.


Teaching–learning-based optimization Modified learner phase Optimal control Nonlinear Volterra integro-differential equation Collocation method Chebyshev wavelet 



The first author acknowledges Gonbad Kavous University, the second author appreciates the Young Researchers and Elite Club, Najafabad Branch, Islamic Azad University, Najafabad, and the third author thanks K.N. Toosi University of Technology for supporting this research work.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • R. Khanduzi
    • 1
    Email author
  • A. Ebrahimzadeh
    • 2
  • M. Reza Peyghami
    • 3
  1. 1.Department of Mathematics and StatisticsGonbad Kavous UniversityGonbad KavousIran
  2. 2.Young Researchers and Elite Club, Najafabad BranchIslamic Azad UniversityNajafabadIran
  3. 3.Faculty of MathematicsK.N. Toosi University of TechnologyTehranIran

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