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An adaptive mesh refinement method considering control errors for pseudospectral discretization

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Listed:
  • Li, Hesong
  • Li, Zhaoting
  • Zhang, Hongbo
  • Wang, Yi

Abstract

This paper presents an adaptive mesh refinement method that considers control errors for solving pseudospectral optimal control problems. Firstly, a method for estimating errors in both states and controls is presented. Based on the estimation results, an adaptive mesh refinement method is subsequently devised. This method increases and reduces the number of collocation points in accordance with a theoretical convergence rate that incorporates both state and control errors. Furthermore, in addition to dividing intervals resulting from a large number of collocation points, new intervals are also generated when control errors exceed tolerance. As a result, the mesh density near the point with the largest control error is effectively increased, thereby improving the discretization accuracy. The effectiveness of the method is illustrated through three numerical examples, and its performance is evaluated in comparison to other adaptive mesh refinement methods. The numerical results demonstrate that the proposed method exhibits superior performance in terms of capturing the nonsmooth and discontinuous changes and achieving an accurate solution, while requiring fewer iterations.

Suggested Citation

  • Li, Hesong & Li, Zhaoting & Zhang, Hongbo & Wang, Yi, 2025. "An adaptive mesh refinement method considering control errors for pseudospectral discretization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 232(C), pages 140-159.
    • Handle: RePEc:eee:matcom:v:232:y:2025:i:c:p:140-159
    DOI: 10.1016/j.matcom.2025.01.005
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    References listed on IDEAS

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    1. Marzban, Hamid Reza, 2022. "A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    2. 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).
    3. Elgindy, Kareem T., 2024. "Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 148-164.
    4. N. Koeppen & I. M. Ross & L. C. Wilcox & R. J. Proulx, 2019. "Fast Mesh Refinement in Pseudospectral Optimal Control," Papers 1904.12992, arXiv.org.
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