球面约束与调和映射流：低模态强迫下的可控性与可达性
Sphere Constraints and Harmonic Map Flow: Controllability and Reachability by Low-Mode Forcing
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中文总结 AI 辅助
研究退化强迫下球面约束演化方程的可控性与可达性，以调和映射热流为例。通过利用几何结构转化为控制仿射系统，分析李代数，证明迭代李括号可产生新方向，为控制流形值演化方程提供李代数框架。
中文摘要 AI 辅助
我们研究退化（低模态）强迫下球面约束演化方程的可控性与可达性，主要应用为调和映射热流。利用潜在几何结构，将问题重新表述为傅里叶变量下的无穷维控制仿射系统，并分析受控向量场生成的李代数。证明迭代李括号产生新的容许方向，为有限多个控制模式在无穷多个傅里叶分量间传播影响提供机制。结果为控制流形值演化方程提供了李代数框架。
英文摘要
We study the controllability and reachability of sphere-constrained evolution equations under degenerate (low-mode) forcing, with the harmonic map heat flow as the principal application. Exploiting the underlying geometric structure, we reformulate the problem as an infinite-dimensional control-affine system in Fourier variables and analyze the Lie algebra generated by the controlled vector fields. We prove that iterated Lie brackets generate new admissible directions, providing a mechanism through which finitely many control modes propagate their influence across infinitely many Fourier components. The results provide a Lie-algebraic framework for controlling manifold-valued evolution equations.
