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

Abstract: 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.