Optimal Control and Geodesics on Quadratic Matrix Lie Groups
Optimal Control and Geodesics on Quadratic Matrix Lie Groups
复制标题
二次矩阵李群的最优控制和测地线
DOI：
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发表时间：
2008
影响因子：
3
通讯作者：
A. Sanyal
中科院分区：
文献类型：
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作者：
A. Bloch;P. Crouch;J. Marsden;A. Sanyal
本文的目的是将刚体方程的对称表示从群SO (N)推广到其他群。这些群是由二次矩阵恒等式定义的一般线性群的矩阵子群。它们对应的李代数包括几个经典的半单矩阵李代数。方法是从这些群上的最优控制问题开始，该问题为左不变度量生成测地线。Bloch，Crouch，Marsden和Ratiu的早期工作定义了刚体方程的对称表示，它是通过在李群So (N)的特殊情况下求解相同的最优控制问题而获得的。本文将这种对称表示推广到满足一定二次矩阵恒等式的更广泛的一类矩阵群。我们考虑了广义刚体方程的这种对称表示与广义刚体方程本身之间的关系。利用这种对称性构造了对称表示的离散化，进而产生了数值算法来积分给定类型的矩阵李群的广义刚体方程。
The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO (n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO (n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.