  In this paper, we introduce the class of diagonally dominant (with respect to
a given LMI region ${\mathfrak D} \subset {\mathbb C}$) matrices that possesses
the analogues of well-known properties of (classical) diagonally dominant
matrices, e.g their spectra are localized inside the region $\mathfrak D$.
Moreover, we show that in some cases, diagonal $\mathfrak D$-dominance implies
$({\mathfrak D}, {\mathcal D})$-stability ( i.e. the preservation of matrix
spectra localization under multiplication by a positive diagonal matrix).
Basing on the properties of diagonal stability and diagonal dominance, we
analyze the conditions for stability of second-order dynamical systems. We show
that these conditions are preserved under system perturbations of a specific
form (so-called $D$-stability). We apply the concept of diagonal $\mathfrak
D$-dominance to the analysis of the minimal decay rate of second-order systems
and its persistence under specific perturbations (so-called relative
$D$-stability). Diagonal $\mathfrak D$-dominance with respect to some conic
region $\mathfrak D$ is also shown to be a sufficient condition for stability
and $D$-stability of fractional-order systems.
