Many real-world decision problems can be naturally modeled as fractional optimal parameter selection and fractional optimal control problems. Therefore, in this paper, we consider a class of combined fractional optimal parameter selection and optimal control problems involving nonlinear fractional systems with Caputo fractional derivatives and subject to canonical equality and inequality constraints. We first approximate this problem by a set of finite-dimensional optimization problems using the control parameterization method, where both the heights of parameterized controls and system parameters are taken as decision variables. We then show that the gradients of the cost and constraint functions with respect to the decision variables can be expressed as the solutions of a series of auxiliary fractional systems, which can be solved together with the original fractional system forward in time, simultaneously. Furthermore, we present a third-order numerical scheme for solving both the original and auxiliary fractional systems. On this basis, a gradient-based optimization algorithm is developed to solve the resulting optimization problems. Finally, we demonstrate the effectiveness and applicability of the developed algorithm through five non-trivial examples, one of which involves the optimal treatment of human immunodeficiency virus.

中文翻译：

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一种求解组合分数阶最优参数选择和最优控制问题的控制参数化方法


许多现实世界的决策问题可以自然地建模为分数阶最优参数选择和分数阶最优控制问题。因此，在本文中，我们考虑了一类组合的分数阶最优参数选择和最优控制问题，这些问题涉及具有 Caputo 分数阶导数的非线性分数系统，并受规范等式和不等式约束。我们首先使用控制参数化方法通过一组有限维优化问题来近似这个问题，其中参数化控制的高度和系统参数都被视为决策变量。然后，我们表明成本和约束函数相对于决策变量的梯度可以表示为一系列辅助分数系统的解，这些解可以与原始分数系统一起在时间上向前求解。此外，我们提出了一种用于求解原始和辅助分数系统（英文）的三阶数值方案。在此基础上，开发了一种基于梯度的优化算法来解决由此产生的优化问题。最后，我们通过五个重要示例证明了所开发算法的有效性和适用性，其中一个示例涉及人类免疫缺陷病毒的最佳治疗。