This paper provides an effective method for a class of 2-dim nonlinear variable order fractional optimal control problems (2DNVOFOCP). The technique is based on the new class of basis functions namely the generalized shifted Legendre polynomials. The dynamic constraint is described by a nonlinear variable order fractional differential equation where the fractional derivative is in the sense of Caputo. The 2-dim Gauss-Legendre quadrature rule together with the Lagrange multipliers method are utilized to find the solutions of the given 2DNVOFOCP. The convergence analysis of the presented method is investigated. The examined numerical examples manifest highly accurate results.

该文为一类2-dim非线性变量阶分数最优控制问题（2DNVOFOCP）提供了一种有效的方法。该技术基于一类新的基础函数，即广义移位的勒让德多项式。动态约束由非线性可变阶分数阶微分方程描述，其中分数导数在Caputo的意义上。2维高斯-勒格德勒正交规则与拉格朗日乘数法一起用于找到给定2DNVOFOCP的解。研究了所提出方法的收敛性分析。检验的数值示例表明结果非常准确。