$\K$-Lorentzian and $\K$-CLC Polynomials in Stability Analysis

TL;DR

This paper explores $ ext{K}$-Lorentzian and $ ext{K}$-CLC polynomials, establishing their stability properties, characterizing conditions for Hurwitz stability, and applying these concepts to analyze the stability of EVI dynamical systems.

Contribution

It introduces an alternative definition of $ ext{K}$-CLC polynomials, characterizes their Hurwitz stability for degrees up to 4 and beyond, and links these polynomials to stability analysis in dynamical systems.

Findings

Strict $ ext{K}$-CLC polynomials of degree ≤ 4 are Hurwitz-stable.

Conditions for Hurwitz stability of degree ≥ 5 $ ext{K}$-CLC polynomials are characterized.

Application of $ ext{K}$-CLC polynomials in stability analysis of EVI systems.

Abstract

We study the class of $\K$-Lorentzian polynomials, a generalization of the distinguished class of Lorentzian polynomials. As shown in \cite{GPlorentzian}, the set of $\K$-Lorentzian polynomials is equivalent to the set of $\K$-completely log-concave (aka $\K$-CLC) forms. Throughout this paper, we interchangeably use the terms $\K$-Lorentzian polynomials for the homogeneous setting and $\K$-CLC polynomials for the non-homogeneous setting. By introducing an alternative definition of $\K$-CLC polynomials through univariate restrictions, we establish that any strictly $\K$-CLC polynomial of degree $d \leq 4$ is Hurwitz-stable polynomial over $\K$. Additionally, we characterize the conditions under which a strictly $\K$-CLC of degree $d \geq 5$ is Hurwitz-stable over $\K$. Furthermore, we associate the largest possible proper cone, denoted by $\K(f,v)$, with a given $\K$-Lorentzian polynomial $f$ in the direction $v \in \inter \K$. Finally, we investigate applications of $\K$-CLC polynomials in the stability analysis of evolution variational inequalities (EVI) dynamical systems governed by differential equations and inequality constraints.