Nonorthogonal polynomials have many useful properties like being used as a basis for spectral methods, being generated in an easy way, having exponential rates of convergence, having fewer terms and reducing computational errors in comparison with some others, and producing most important basic polynomials. In this regard, this paper deals with a new indirect numerical method to solve fractional optimal control problems based on the generalized Lucas polynomials. Through the way, the left and right Caputo fractional derivatives operational matrices for these polynomials are derived. Based on the Pontryagin maximum principle, the necessary optimality conditions for this problem reduce into a two-point boundary value problem. The main and efficient characteristic behind the proposed method is to convert the problem under consideration into a system of algebraic equations which reduces many computational costs and CPU time. To demonstrate the efficiency, applicability, and simplicity of the proposed method, several examples are solved, and the obtained results are compared with those obtained with other methods.

非正交多项式具有许多有用的特性，例如用作谱方法的基础、以简单的方式生成、具有指数收敛率、与其他多项式相比具有更少的项和减少计算错误，以及产生最重要的基本多项式。在这方面，本文提出了一种新的间接数值方法来解决基于广义卢卡斯多项式的分数阶最优控制问题。顺便推导出这些多项式的左右Caputo分数阶导数运算矩阵。基于Pontryagin极大值原理，该问题的必要最优性条件简化为两点边值问题。所提出方法背后的主要和有效特征是将所考虑的问题转换为代数方程组，从而减少许多计算成本和 CPU 时间。为了证明所提出方法的效率、适用性和简单性，解决了几个例子，并将获得的结果与其他方法获得的结果进行了比较。