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

TL;DR

This paper introduces a generalized class of diagonally dominant matrices relative to an LMI region, exploring their spectral properties, stability implications for second-order and fractional systems, and robustness under perturbations.

Contribution

It extends classical diagonal dominance concepts to LMI regions, establishing new stability criteria and analyzing their effects on system decay rates and perturbation resilience.

Findings

Diagonal $rak D$-dominance localizes spectra within $rak D$

Diagonal $rak D$-dominance implies $({rak D},{rak D})$-stability in some cases

Conditions for stability are preserved under specific perturbations

Abstract

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.