This paper presents an efficient operational matrix method for two dimensional nonlinear variable order fractional optimal control problems (2D-NVOFOCP). These problems include the nonlinear variable order fractional dynamical systems (NVOFDS) described by partial differential equations such as the diffusion-wave, convection-diffusion-wave and Klein-Gordon equations. The variable order fractional derivative is defined in the Caputo type. The proposed hybrid method is based on the transcendental Bernstein series (TBS) and the generalized shifted Chebyshev polynomials (GSCP). The new operational matrices of derivatives are generated for the mentioned polynomials. The state and control functions are expressed by the TBS and GSCP with free coefficients and control parameters. These expansions are substituted in the performance index and the resulting operational matrices are employed to extract algebraic equations from the approximated NVOFDS. The constrained extremum is obtained by coupling the algebraic constraints from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the performance index by means of a set of unknown Lagrange multipliers. The convergence analysis is discussed and several numerical experiments illustrate the efficiency and accuracy of the proposed method.

中文翻译：

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二维非线性变阶分数最优控制问题的一种新的混合方法

本文提出了一种用于二维非线性变阶分数最优控制问题 (2D-NVOFOCP) 的有效运算矩阵方法。这些问题包括由偏微分方程（如扩散波、对流扩散波和 Klein-Gordon 方程）描述的非线性可变阶分数动力系统 (NVOFDS)。可变阶分数导数在 Caputo 类型中定义。所提出的混合方法基于超越伯恩斯坦级数 (TBS) 和广义平移切比雪夫多项式 (GSCP)。为上述多项式生成新的导数运算矩阵。状态和控制函数由带有自由系数和控制参数的 TBS 和 GSCP 表示。这些扩展被替换在性能指标中，并且产生的运算矩阵用于从近似的 NVOFDS 中提取代数方程。约束极值是通过将来自动力系统和初始和边界条件的代数约束与通过一组未知的拉格朗日乘子从性能指标中提取的代数方程耦合来获得的。讨论了收敛性分析，并通过几个数值实验说明了所提出方法的效率和准确性。约束极值是通过将来自动力系统和初始和边界条件的代数约束与通过一组未知的拉格朗日乘子从性能指标中提取的代数方程耦合来获得的。讨论了收敛性分析，并通过几个数值实验说明了所提出方法的效率和准确性。约束极值是通过将来自动力系统的代数约束以及初始和边界条件与通过一组未知的拉格朗日乘子从性能指标中提取的代数方程耦合来获得的。讨论了收敛性分析，并通过几个数值实验说明了所提出方法的效率和准确性。