分数阶线性系统终端时刻固定的最优控制问题
Optimal Control Problems with a Fixed Terminal Time in Linear Fractional-Order Systems
摘要：
研究了一类由具有Caputo分数阶导数的线性微分方程描述的动力系统的最优控制问题。控制的目标是最小化Bolza型成本泛函，它由两项组成：第一项评估系统在固定终端时间的状态，第二项是整个时间间隔上的控制的积分评估。为了解决这个问题，我们建议把它归结为一个由一阶常微分方程描述的动力系统中的一些辅助最优控制问题。该约化是基于线性分数阶微分方程解的表示公式，并通过某种线性变换来实现的，它被称为原系统位置的信息图像，可以被看作是该系统在终端时刻运动的一种特殊预测。原始和辅助问题之间的连接建立开环和反馈（闭环）控制。文中所得结果用实例加以说明。
英文摘要：
The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.
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