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<p>This paper presents a reliable numerical approach based on Legendre polynomials for solving a class of three-dimensional fractional optimal control problems (3D-FOCPs). The fractional derivative is expressed in the Caputo sense. The proposed method employs the Ritz technique to approximate the state and control functions, offering significant flexibility in handling both the initial and boundary conditions associated with 3D-FOCPs. The original problem is converted into a system of algebraic equations by utilizing Legendre polynomials in the Ritz framework. The convergence properties of the proposed method are thoroughly analyzed. To validate the effectiveness and applicability of the method, two non-trivial examples are presented with varying values of the fractional-order parameter, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. The numerical results are compared with those obtained using the eigenfunction method. Notably, the results demonstrate that as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> approaches unity, a small number of Legendre polynomial terms is sufficient to recover the semi-analytical solutions derived via the eigenfunction method. The numerical and graphical results highlight the accuracy, efficiency, and reliability of the proposed approach, underscoring its potential for effectively solving complex fractional optimal control problems.</p>

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A Novel Numerical Approach for 3D Fractional Optimal Control Problems Using Legendre Polynomials

  • 期刊论文

  • 发表日期：

    2025年11月19日

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  • Kamal Mamehrashi

摘要

This paper presents a reliable numerical approach based on Legendre polynomials for solving a class of three-dimensional fractional optimal control problems (3D-FOCPs). The fractional derivative is expressed in the Caputo sense. The proposed method employs the Ritz technique to approximate the state and control functions, offering significant flexibility in handling both the initial and boundary conditions associated with 3D-FOCPs. The original problem is converted into a system of algebraic equations by utilizing Legendre polynomials in the Ritz framework. The convergence properties of the proposed method are thoroughly analyzed. To validate the effectiveness and applicability of the method, two non-trivial examples are presented with varying values of the fractional-order parameter, \(\alpha \) α . The numerical results are compared with those obtained using the eigenfunction method. Notably, the results demonstrate that as \(\alpha \) α approaches unity, a small number of Legendre polynomial terms is sufficient to recover the semi-analytical solutions derived via the eigenfunction method. The numerical and graphical results highlight the accuracy, efficiency, and reliability of the proposed approach, underscoring its potential for effectively solving complex fractional optimal control problems.