Optimal control of fractional integro-differential systems based on a spectral method and grey wolf optimizer





Optimal control, Fractional Volterra integro-differential equation, collocation method, Grey wolf optimizer


This paper elaborated an effective and robust metaheuristic algorithm with acceptable performance based on solution accuracy. The algorithm applied in solution of the optimal control of fractional Volterra integro-differential (FVID) equation which be substituted by nonlinear programming (NLP). Subsequently the FIVD convert the problem to a NLP by using spectral collocation techniques and thereafter we execute the grey wolf optimizer (GWO) to improve the speed and accuracy and find the solutions of the optimal control and state as well as the optimal value of the cost function. It is mentioned that the utilization of the GWO is simple, due to the fact that the GWO is global search algorithm, the method can be applied to find optimal solution of the NLP. The efficiency of the proposed scheme is shown by the results obtained in comparison with the local methods. Further, some illustrative examples introduced with their approximate solutions and the results of the present approach compared with those achieved using other methods.


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Diethelm, K., Ford, N.J. and Freed, A.D. (2004). Detailed error analysis for a fractional Adams method. Numerical Algorithms,36, 31–52.

Huang, L., Li, X.F., Zhao, Y.L. and Duan, X.Y. (2011). Approximate solution of fractional integro-differential equations by Taylor expansion method. Computers & Mathematics with Applications, 62(3), 1127–1134.

Tohidi, E. and Nik, H.S. (2015). A bessel collocation method for solving fractional optimal control problems. Applied Mathematical Modelling, 39(2), 455–465.

Keshavarz, E., Ordokhani, Y. and Razzaghi, M. (2015). A numerical solution for fractional optimal control problems via Bernoulli polynomials. Journal of Vibration and Control, 22(18), 3889–3903.

Zaky, M.A. and Machado, J.A.T. (2017). On the formulation and numerical simulation of distributed-order fractional optimal control problems. Communications in Nonlinear Science and Numerical Simulation, 52, 177–189.

Salati, A.B., Shamsi, M., Torres, D.F.M. (2019). Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems. Communications in Nonlinear Science and Numerical Simulation, 67, 334–350.

Youssri, Y.H., and Abd-Elhameed, W.M. (2018). Spectral tau algorithm for solving a class of fractional optimal control problems via Jacobi polynomials. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(2), 152–160.

Youssri, Y.H. and Hafez, R.M. (2019). Chebyshev collocation treatment of Volterra Fredholm integral equation with error analysis. Arabian Journal of Mathematics, 1–10.

Mirjalili, S.A., Mirjalili, S.M. and Lewis, A. (2014). Grey wolf optimizer. Advances in Engineering Software, 69, 46–61.

Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012). Fractional calculus models and numerical methods. Nonlinearity and Chaos, Series on Complexity, World Scientific.

Costabile, F., Dellaccio, F. and Gualtieri, M.I. (2006). A new approach to Bernoulli polynomials. Rendiconti di Matematica, Serie VII, 26, 1–12.

Arfken G. (1985). Mathematical methods for physicists. 3rd edn. San Diego, CA: Academic Press.

Kreyszig, E. (1978). Introductory functional analysis with applications. New York: John Wiley and Sons.

Yuan, Y.X. (2000). A review of trust region algorithms for optimization. In Iciam, 99(1), 271–282.

Sadjadi, S.J. and Ponnambalam, K. (1999). Advances in trust region algorithms for constrained optimization. Applied Numerical Mathematics, 29(3), 423–443.

Faris, H., Aljarah, I., Al-Betar, M. A. and Mirjalili, S. (2018). Grey wolf optimizer: a review of recent variants and applications. Neural computing and applications, 30(2), 413–435.

Wang, J.S. and Li, S.X. (2019). An improved grey wolf optimizer based on differential evolution and elimination mechanism. Scientific reports, 9(1), 71–81.

Liu, H., Hua, G., Yin, H. and Xu, Y. (2018). An intelligent grey wolf optimizer algorithm for distributed compressed sensing. Computational intelligence and neuroscience, 2018.

Gupta, S. and Deep, K. (2018). An opposition-based chaotic grey wolf optimizer for global optimisation tasks. Journal of Experimental & Theoretical Artificial Intelligence, 1–29.

Abdo, M., Kamel, S., Ebeed, M., Yu, J. and Jurado, F. (2018). Solving non-smooth optimal power flow problems using a developed grey wolf optimizer. Energies, 11(7), 1692.

Pradhan, M., Roy, P. K. and Pal, T. (2016). Grey wolf optimization applied to economic load dispatch problems. International Journal of Electrical Power & Energy Systems, 83, 325–334.

Tung, N.S. and Chakravorty, S. (2015). Grey Wolf optimization for active power dispatch planning problem considering generator constraints and valve point effect. International Journal of Hybrid Information Technology, 8(12), 117–134.

Maleknejad, K., Ebrahimzadeh, A. (2014). Optimal control of volterra integro- differential systems based on legendre wavelets and collocation method. Journal of Mathematical, Computational, Physical an Quantum Engineering,8(7), 1040–1044.



DOI: 10.11121/ijocta.01.2020.00753
Published: 2020-01-14

How to Cite

Khanduzi, R., Ebrahimzadeh, A., & Panjeh Ali Beik, S. (2020). Optimal control of fractional integro-differential systems based on a spectral method and grey wolf optimizer. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10(1), 55–65. https://doi.org/10.11121/ijocta.01.2020.00753



Research Articles