In this paper, we study optimal control problems for nonlinear fractional order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the $ L2 $ scheme and the Grünwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are provided to demonstrate the feasibility of the numerical method.

中文翻译：

---

星图上的分数最优控制问题：最优系统和数值解

在本文中，我们研究星形图上非线性分数阶边值问题的最优控制问题，其中分数导数以Caputo形式描述。通过拉格朗日乘数法，导出了分数最优控制问题（FOCP）的伴随状态和最优系统。然后，利用Banach压缩原理证明了伴随方程解的存在性和唯一性。我们还提出了一种数值方法来找到所得最优系统的近似解。在提出的方法中，使用$ L2 $方案和Grünwald-Letnikov公式分别对Caputo分数导数和右Riemann-Liouville分数导数进行逼近，将最优系统转换为线性代数方程组。