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Analysis of nonlinear fractional optimal control systems described by delay Volterra–Fredholm integral equations via a new spectral collocation method

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  • Marzban, Hamid Reza
  • Nezami, Atiyeh

Abstract

We aim to introduce a new spectral collocation method for investigating and analyzing nonlinear delay control systems governed by the fractional mixed Volterra–Fredholm integral equations (MVFIEs). The generalized fractional Legendre basis (GFLB) is used as a complete orthogonal basis, and the fractional Legendre–Gaussian nodes are introduced and employed as the fractional collocation points. These nodes correspond to the zeros of the fractional-order Legendre function of degree M. The convergence of the generalized orthogonal basis is discussed in detail based on the Sobolev and L2 norms. Additionally, new fractional integral and delay operators are implemented to reduce the primary control system to a static optimization system. Two benchmark fractional control problems, including delay, are considered to demonstrate the powerfulness and superiority of the new methodology. The introduced collocation approach can be successfully performed for solving even those fractional control problems having any irregularities in the control function, including jump discontinuities and bang–bang behavior.

Suggested Citation

  • Marzban, Hamid Reza & Nezami, Atiyeh, 2022. "Analysis of nonlinear fractional optimal control systems described by delay Volterra–Fredholm integral equations via a new spectral collocation method," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    • Handle: RePEc:eee:chsofr:v:162:y:2022:i:c:s0960077922007044
    DOI: 10.1016/j.chaos.2022.112499
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    References listed on IDEAS

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    1. Chongyang Liu & Zhaohua Gong & Changjun Yu & Song Wang & Kok Lay Teo, 2021. "Optimal Control Computation for Nonlinear Fractional Time-Delay Systems with State Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 83-117, October.
    2. A. H. Bhrawy & L. M. Assas & M. A. Alghamdi, 2013. "Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, August.
    3. Heydari, M.H. & Razzaghi, M., 2021. "Piecewise Chebyshev cardinal functions: Application for constrained fractional optimal control problems," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    4. Heydari, M.H. & Razzaghi, M., 2021. "A numerical approach for a class of nonlinear optimal control problems with piecewise fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Haniye Dehestani & Yadollah Ordokhani & Mohsen Razzaghi, 2020. "Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation error," International Journal of Systems Science, Taylor & Francis Journals, vol. 51(6), pages 1032-1052, April.
    6. Changjun Yu & Qun Lin & Ryan Loxton & Kok Lay Teo & Guoqiang Wang, 2016. "A Hybrid Time-Scaling Transformation for Time-Delay Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 876-901, June.
    7. Borah, Manashita & Das, Debanita & Gayan, Antara & Fenton, Flavio & Cherry, Elizabeth, 2021. "Control and anticontrol of chaos in fractional-order models of Diabetes, HIV, Dengue, Migraine, Parkinson's and Ebola virus diseases," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    8. Haifa Bin Jebreen & Ioannis Dassios, 2022. "On the Wavelet Collocation Method for Solving Fractional Fredholm Integro-Differential Equations," Mathematics, MDPI, vol. 10(8), pages 1-12, April.
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    3. J., Kokila & M., Vellappandi & D., Meghana & V., Govindaraj, 2023. "Optimal control study on Michaelis–Menten kinetics — A fractional version," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 571-592.

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