Many significant real world challenges arise as optimization problems on different classes of control systems. In particular, ordinary differential equations with symmetries. The purpose of this review article is twofold. First, we give the information we have about the class of Linear Control Systems ${\mathsf{\Sigma }}_G$ on a low dimension matrix Lie group G. Second, we invite the Mathematical community to consider possible applications through the Pontryagin Maximum Principle for ${\mathsf{\Sigma }}_G$. In addition, we challenge some theoretical open problems. The class ${\mathsf{\Sigma }}_G$ is a perfect generalization of the classical Linear Control System on the Abelian group ${\mathbb{R}}^n$. Let G be a Lie group of dimension two or three. Related to ${\mathsf{\Sigma }}_G$, this review describes the actual results about controllability, the time-optimal Hamiltonian equations and, the Pontryagin Maximum Principle. We show how to build ${\mathsf{\Sigma }}_G$, through several examples on low dimensional matrix groups.

中文翻译：

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线性控制系统在现实世界中的应用和理论挑战

随着不同类别的控制系统上的优化问题，现实世界中出现了许多重大挑战。特别是具有对称性的常微分方程。这篇评论文章的目的是双重的。首先，我们提供有关线性控制系统类别的信息${\mathsf{\Sigma }}_G$上的低维矩阵李群ģ。其次，我们邀请数学界考虑通过Pontryagin极大原理计算的可能应用${\mathsf{\Sigma }}_G$。另外，我们挑战一些理论上的开放性问题。班级${\mathsf{\Sigma }}_G$ 是经典线性控制系统在Abelian组上的完美概括 ${\mathbb{[R}}^{ñ}$。令G为二维或三维的李群。相关${\mathsf{\Sigma }}_G$，这篇综述描述了关于可控性，时间最优哈密顿方程和庞特里亚金最大原理的实际结果。我们展示了如何建造${\mathsf{\Sigma }}_G$，通过有关低维矩阵组的几个示例。