论文标题
Lie Systems的几何数值方法及其在最佳控制中的应用
Geometric numerical methods for Lie systems and their application in optimal control
论文作者
论文摘要
Lie System是一阶普通微分方程的非自主系统，可以通过自主函数编写一般解决方案，这是一个有限数量的特定解决方案的所谓（非线性）叠加规则，某些参数与初始条件有关。即使已知某些谎言系统的叠加规则，其解决方案的明确分析表达也不是。这就是为什么本文重点介绍了一种新的几何尝试，以分析和数值整合谎言系统的原因。我们专注于两个方法家族：基于Magnus扩展和Runge-Kutta-Munthe-Kaas方法的家族，这些方法适用于Lie Systems的几何特性。为了说明我们的技术的准确性，我们根据SL $（N，\ Mathbb {r}）$ Lie Group提出了示例，该组在力学中起着非常相关的作用。特别是，我们描绘了具有二次成本功能的车辆的最佳控制问题。给出了研究示例的特定数值解决方案。
A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, a so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. Even if the superposition rules for some Lie systems are known, the explicit analytic expression of their solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods: those based on Magnus expansions and the Runge-Kutta-Munthe-Kaas method, which are here adapted to the geometric properties of Lie systems. To illustrate the accuracy of our techniques we propose examples based on the SL$(n,\mathbb{R})$ Lie group, which plays a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.
