This work develops an optimization method based on a new class of basis function, namely the generalized Bernoulli polynomials (GBP), to solve a class of nonlinear 2-dim fractional optimal control problems. The problem is generated by nonlinear fractional dynamical systems with fractional derivative in the Caputo type and the Goursat–Darboux conditions. First, we use the GBP to approximate the state and control variables with unknown coefficients and parameters. Afterwards, we substitute the obtained values for the variables and parameters in the objective function, nonlinear fractional dynamical system and Goursat–Darboux conditions. The 2-dim Gauss–Legendre quadrature rule together with a fractional operational matrix construct a constrained problem, that is solved by the Lagrange multipliers method. The convergence of the GBP method is proved and its efficiency is demonstrated by several examples.

这项工作开发了一种基于一类新的基础函数的优化方法，即广义伯努利多项式（GBP），以解决一类非线性2维分数分数最优控制问题。该问题是由非线性分数动力系统产生的，该系统具有Caputo类型和Goursat–Darboux条件下的分数导数。首先，我们使用GBP来估计状态和具有未知系数和参数的控制变量。然后，我们用获得的值代替目标函数，非线性分数动力系统和Goursat–Darboux条件中的变量和参数。2维高斯-勒格德勒正交规则与分数运算矩阵共同构成了一个受约束的问题，该问题可以通过拉格朗日乘数法来解决。