A Novel Approach to D-Stability and Additive D-Stability of Economic Models
Abstract
The study of \(D\)-stability in mathematical analysis is crucial for understanding and ensuring the stability of linear dynamical systems. This article introduces novel findings on the characterization of \(D\)-stability, along with its connections to additive \(D\)-stability concerning speed and coordinate transformations in linear dynamical systems with \(n\) degrees of freedom\[A \frac{d^2\mu(\tau)}{d\tau^2} + B \frac{d\mu(\tau)}{d\tau} + C \mu(\tau) = 0, \ \tau \in \mathbb{R}, \ \tau > 0,\]Consider the stiffness, mass, and damping matrices \(A, B, C \in \mathcal{M}^{n \times n}\), and let \( \mu(\tau) \in \mathbb{R}^n \) denote the vector of generalized coordinates with \(\frac{d\mu(\tau)}{d\tau}\) representing its corresponding velocity vector. This work derives new theoretical insights into \(D\)-stability, additive \(D\)-stability with respect to velocity, and additive \(D\)-stability concerning coordinate transformations. These results are established using techniques from linear algebra, matrix theory, dynamical systems, and their connections to structured singular value computations. Additionally, numerical investigations of the spectrum, singular values, and pseudospectra of the coefficient matrices \(A, B, C \in \mathcal{M}^{n \times n}\) are conducted using EigTool, providing further validation of the theoretical framework.