Asymptotic stability, $ \mathfrak{D} $-stability, strong $ \mathfrak{D} $-stability, $ \hat{H} $-stability, and $ \mathfrak{D}(\alpha) $-stability

In this paper, we analyze the stability of linear interval matrix systems in the presence of parametric uncertainties. We aim to develop mathematical conditions which allow different notions of stability within a unified framework based on structured singular values. In particular, we aim to establish both necessary and sufficient criteria for asymptotic stability, $ \mathfrak{D} $-stability, strong $ \mathfrak{D} $-stability, $ \hat{H} $-stability, and $ \mathfrak{D}(\alpha) $-stability. The proposed formulation can be viewed as an extension to classical diagonal stability with interval uncertainty, where the system parameters vary within prescribed bounds. The main results depend on vertex-based analysis combined with structured perturbation techniques and provide a way to quantify robustness margins. To illustrate the applicability of the proposed methodology, several numerical examples are presented, including cases motivated by aerospace and industrial systems. The results indicate that the proposed framework offers a noticeable reduction in computational effort while preserving the theoretical properties of existing methods.

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