AI 中文总结
本文构建李理论框架，将线性可控性拓展至矩阵李群的非线性可控性，以旋转群为重点，结合仿真与理论对比，研究局部信息向可达集的传播机制。
AI 中文摘要
本文构建了一套自洽的李理论框架，用于研究从线性可控性到矩阵李群上非线性可控性的过渡，核心问题是局部代数信息能否转化为可达集的相关结论。线性情形提供了参照模型：矩阵指数、凯莱-哈密顿定理、轨迹公式及卡尔曼族{B,AB,…,A^{n-1}B}将可控性问题简化为有限维线性代数问题。在非线性情形中，李括号取代了矩阵幂，但对应的李代数可能为无限维，有限维李对应关系可能失效。研究特别关注旋转群：反对称矩阵、其指数、交换子、单参数子群及贝克-坎贝尔-豪斯多夫公式，为观察SO(n)上无穷小方向的传播提供了具体研究对象。数值SO(3)仿真展示了受控旋转的几何特性，明确的SO(4)与SO(7)交换子序列则使局部控制方向的传播可视化。本文随后为右不变系统构建了可达子群论证，给出了雅姆贝步背后的详细证明序列，并针对上三角矩阵代数的可解、幂零及理想示例展开研究。最后将该有限维机制与Sussmann的一般非线性系统局部可控性理论进行对比。
英文摘要
This article develops a self-contained Lie-theoretic route from linear controllability to nonlinear controllability on matrix Lie groups. The organizing question is whether local algebraic information can be propagated into statements about reachable sets. The linear case supplies the model: the matrix exponential, Cayley-Hamilton theorem, trajectory formula and Kalman family B,AB,...,An-1B reduce controllability to finite-dimensional linear algebra. In the nonlinear setting, Lie brackets replace matrix powers, but the associated Lie algebra may be infinite dimensional and the finite-dimensional Lie correspondence can fail. Particular emphasis is placed on rotation groups. Skew-symmetric matrices, their exponentials, commutators, one-parameter subgroups, and the Baker-Campbell-Hausdorff formula provide a concrete laboratory for seeing how infinitesimal directions propagate on SO(n). A numerical SO(3) simulation illustrates the geometry of controlled rotations, while explicit SO(4) and SO(7) commutator sequences make the propagation of a localized control direction visible. The article then develops the attainable-subgroup argument for right-invariant systems, gives a detailed proof sequence behind the Yamabe step, and works through solvable, nilpotent and ideal examples for upper-triangular matrix algebras. The finite-dimensional mechanism is finally contrasted with Sussmann's local controllability theory for general nonlinear systems.
CommentsNo original claim in this paper. It is a pedagogical exposition giving important links to the literature, espcially with Yamabe Theorem on arcwise connected subgroups
