Generalized D-stability and diagonal dominance with applications to stability and transient response properties of systems of ODE

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.