We consider an optimal control problem for a dynamical system whose motion is described by a linear differential equation with the Caputo fractional derivative of order \(\alpha\in(0,1)\). The time interval of the control process is fixed and finite. The control actions are subject to geometric constraints. The aim of the control is to minimize a given terminal-integral performance index. In order to construct a solution, we develop the following approach. First, from the considered problem, we turn to an auxiliary optimal control problem for a first-order linear system with lumped delays, which approximates the original system. After that, the auxiliary problem is reduced to an optimal control problem for an ordinary differential system. Based on this, we propose a closed-loop scheme of optimal control of the original system that uses the approximating system as a guide. In this scheme, the control in the approximating system is formed with the help of an optimal positional control strategy from the reduced problem. The effectiveness of the developed approach is illustrated by a problem in which the performance index is the norm of the terminal state of the system.

我们考虑动态系统的最优控制问题，其运动由线性微分方程描述，其阶数为\(\alpha\in(0,1)\). 控制过程的时间间隔是固定的和有限的。控制动作受几何约束。控制的目的是最小化给定的终端综合性能指数。为了构建解决方案，我们开发了以下方法。首先，从所考虑的问题，我们转向具有集中延迟的一阶线性系统的辅助最优控制问题，它近似于原始系统。之后，将辅助问题简化为普通微分系统的最优控制问题。在此基础上，我们提出了一种以逼近系统为指导的原始系统最优控制闭环方案。在该方案中，逼近系统中的控制是在简化问题的最佳位置控制策略的帮助下形成的。