标题： 从可控性到信息：通过格拉姆矩阵、最小能量、费舍尔信息矩阵和熵的统一分析 显示英文标题

标题： From Controllability to Information: A Unified Analysis via Gramian, Minimum Energy, Fisher Information Matrix and Entropy

摘要： 本文探讨了线性动力系统背景下可控性、控制能量、信息和熵之间的联系，以阻尼谐波振子为例进行研究。 我们解析推导了不同阻尼区域下的可控性Gram矩阵（Wc）及其行列式（det(Wc)）。 我们从最小控制能量的角度分析了det(Wc)的物理意义，强调了系统参数（阻尼{\zeta }和自然频率{\omega }n）的影响。 我们研究了Gram矩阵、Fisher信息矩阵（I）以及Shannon（H）和热力学（S）熵之间的概念和形式关系，提出了能量控制与估计精度之间的对偶性，以及控制难易程度与信息/热力学熵之间的联系。 统一的分析为支配动力系统控制的基本原理提供了见解，整合了能量和信息视角。 显示英文摘要

摘要： This article explores the connections between controllability, control energy, information, and entropy in the context of linear dynamical systems, using the damped harmonic oscillator as a case study. We analytically derive the Controllability Gramian Matrix (Wc) and its determinant (det(Wc)) for different damping regimes. We analyze the physical interpretation of det(Wc) in terms of minimum control energy, highlighting the influence of system parameters (damping {\zeta} and natural frequency {\omega}n). We investigate the conceptual and formal relationships between the Gramian, the Fisher Information Matrix (I), and Shannon (H) and thermodynamic (S) entropies, suggesting a duality between energetic control and estimation precision, as well as a link between ease of control and informational/thermodynamic entropy. The unified analysis offers insights into the fundamental principles governing the control of dynamical systems, integrating energy and information perspectives.