广义欧拉角的最优估计及其在经典和量子控制中的应用
Optimal Evaluation of Generalized Euler Angles with Applications to Classical and Quantum Control
摘要：
给定$so(3)$中两个线性无关的矩阵$Z_1$和$Z_2$，每个旋转矩阵$X_f \in SO(3)$可以被写为来自对应于$Z_1$和$Z_2$的一维子群的交错元素的乘积，即$X_f=e^{Z_1 t_1}e^{Z_2 t_2}e^{Z_1t_3} \cdot \cdot \cdot e^{Z_1t_s}$。参数$t_i$、$i=1,...,s$称为{ \it广义欧拉角}。在本文中，我们评估的最小数量的因子所需的因式分解的$X_f \in SO(3)$，作为一个功能的$X_f$，并提供了一个算法来确定广义欧拉角显式。所得结果可应用于某些经典控制系统和二能级量子系统的开关数最少的bang-bang控制。
英文摘要：
Given two linearly independent matrices in $so(3)$, $Z_1$ and $Z_2$, every rotation matrix $X_f \in SO(3)$ can be written as the product of alternate elements from the one dimensional subgroups corresponding to $Z_1$ and $Z_2$, namely $X_f=e^{Z_1 t_1}e^{Z_2 t_2}e^{Z_1t_3} \cdot \cdot \cdot e^{Z_1t_s}$. The parameters $t_i$, $i=1,...,s$ are called {\it generalized Euler angles}. In this paper, we evaluate the minimum number of factors required for the factorization of $X_f \in SO(3)$, as a function of $X_f$, and provide an algorithm to determine the generalized Euler angles explicitly. The results can be applied to the bang bang control with minimum number of switches of some classical control systems and of two level quantum systems.
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