The main purpose of this work is to develop a numerical solution method for solving a class of nonlinear free final time fractional optimal control problems. This problem is subject to equality and inequality constraints in canonical forms, and the orders in the fractional system can be different. For this problem, we first show that, by a time-scaling transformation, the problem can be transformed into an equivalent fractional optimal control problem with fixed final time. We then discretize the transformed fractional optimal control problem by a second-order one-point numerical integration scheme and the trapezoidal rule. Furthermore, we derive the gradient formulae of the cost and constraint functions with respect to decision variables and propose a numerical procedure for calculating these gradients. On this basis, a gradient-based optimization algorithm is developed for solving the resulting problem. Finally, numerical simulations of three example problems illustrate the effectiveness of the developed algorithm.

这项工作的主要目的是开发一种求解一类非线性自由最终时间分数最优控制问题的数值求解方法。此问题受到规范形式的等式和不等式约束，分数系统中的阶数可以不同。对于这个问题，我们首先表明，通过时间标度转换，可以将该问题转换为最终时间固定的等效分数最优控制问题。然后，通过二阶一点数值积分方案和梯形法则离散化变换的分数最优控制问题。此外，我们针对决策变量推导了成本函数和约束函数的梯度公式，并提出了计算这些梯度的数值程序。以这个为基础，开发了基于梯度的优化算法来解决所产生的问题。最后，对三个示例问题的数值模拟说明了所开发算法的有效性。