A physics-informed Chebyshev spectral neural network for distributed-order fractional optimal control problems

Ishtiaq Ali; Ishtiaq Ali

doi:10.3934/math.2026940

AIMS Mathematics

2026, Volume 11, Issue 7: 23308-23343. doi: 10.3934/math.2026940

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A physics-informed Chebyshev spectral neural network for distributed-order fractional optimal control problems

Ishtiaq Ali ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 12 May 2026 Revised: 21 July 2026 Accepted: 22 July 2026 Published: 31 July 2026
MSC : 26A33, 49M25, 65L60

In this paper, a physics-informed Chebyshev spectral neural network (PI-CSNN) framework was proposed for solving distributed-order fractional optimal control problems (DOFOCPs). The proposed approach combines Chebyshev spectral approximations with physics-informed residual minimization to construct an accurate computational framework for DOFOCPs. The distributed-order fractional operators are incorporated directly into the neural-spectral formulation, allowing the method to treat both single-order and distributed-order optimal control models within a unified framework. Several numerical examples involving solutions with limited regularity and time-invariant and time-varying DOFOCPs were presented to evaluate the performance of the proposed method. The numerical results demonstrated high accuracy, systematic convergence, and reliable approximations of the state, adjoint, and control variables. Comparisons with existing numerical methods showed that the proposed PI-CSNN framework achieves competitive optimal cost values and close agreement with benchmark solutions. Overall, the proposed PI-CSNN framework provides an accurate and flexible computational methodology for DOFOCPs.
Citation: Ishtiaq Ali. A physics-informed Chebyshev spectral neural network for distributed-order fractional optimal control problems[J]. AIMS Mathematics, 2026, 11(7): 23308-23343. doi: 10.3934/math.2026940

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Abstract

In this paper, a physics-informed Chebyshev spectral neural network (PI-CSNN) framework was proposed for solving distributed-order fractional optimal control problems (DOFOCPs). The proposed approach combines Chebyshev spectral approximations with physics-informed residual minimization to construct an accurate computational framework for DOFOCPs. The distributed-order fractional operators are incorporated directly into the neural-spectral formulation, allowing the method to treat both single-order and distributed-order optimal control models within a unified framework. Several numerical examples involving solutions with limited regularity and time-invariant and time-varying DOFOCPs were presented to evaluate the performance of the proposed method. The numerical results demonstrated high accuracy, systematic convergence, and reliable approximations of the state, adjoint, and control variables. Comparisons with existing numerical methods showed that the proposed PI-CSNN framework achieves competitive optimal cost values and close agreement with benchmark solutions. Overall, the proposed PI-CSNN framework provides an accurate and flexible computational methodology for DOFOCPs.

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