In this manuscript, we discuss the optimal control problem for a nonlinear system governed by the fractional differential equation in a separable Hilbert space $ X $. We utilize the fixed point technique and $ \eta $-resolvent family to present the existence of control for the fractional system. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem. Further, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Finally, the main results are validated with the aid of an example.

中文翻译：

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分数阶微分方程控制性问题的逼近定理

在本手稿中，我们讨论了可分希尔伯特空间$ X $中由分数阶微分方程控制的非线性系统的最优控制问题。我们利用定点技术和$ \ eta $ -resolvent系列介绍分数系统的控制权存在。获得最优对作为无约束问题的最优对序列的极限。此外，我们得出一些近似结果，从而保证了数值方法收敛于最优对序列。最后，通过一个例子验证了主要结果。