Invariance of Hurwitz-Stability of Polynomials of Degree Five under Positive Hadamard Powers
Abstract
A complete characterization of the stability-preserving exponent set for fractional Hadamard powers of a monic Hurwitz-stable polynomial f of degree five is presented. The stability problem is reduced to the analysis of a single scalar function depending only on three parameters formed from the coefficients of the polynomial f. This reduction leads to a unique stability threshold p*(f) ∈ (0,1) such that the p-th Hadamard power of f is Hurwitz stable if and only if p > p*(f). Consequently, the stability-preserving exponent set is precisely (p*(f),∞). This threshold depends smoothly on the parameters and provides a global coordinate on the admissible parameter region. Finally, smooth dependence is illustrated by a one-parameter family of Hurwitz-stable polynomials whose complex-conjugate zeros approach the imaginary axis.
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BibTeX
@misc{alseidi2026invariance,
title = {{Invariance of Hurwitz-Stability of Polynomials of Degree Five under Positive Hadamard Powers}},
author = {Rola Alseidi and Raoufah Alsaidi and Jürgen Garloff},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16920},
eprint = {2609.16920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16920},
}
RIS
TY - UNPB
TI - Invariance of Hurwitz-Stability of Polynomials of Degree Five under Positive Hadamard Powers
AU - Rola Alseidi
AU - Raoufah Alsaidi
AU - Jürgen Garloff
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16920
UR - https://arxiv.org/abs/2609.16920
AB - A complete characterization of the stability-preserving exponent set for fractional Hadamard powers of a monic Hurwitz-stable polynomial f of degree five is presented. The stability problem is reduced to the analysis of a single scalar function depending only on three parameters formed from the coefficients of the polynomial f. This reduction leads to a unique stability threshold p_*(f) ∈ (0,1) such that the p-th Hadamard power of f is Hurwitz stable if and only if p > p_*(f). Consequently, the stability-preserving exponent set is precisely (p_*(f),∞). This threshold depends smoothly on the parameters and provides a global coordinate on the admissible parameter region. Finally, smooth dependence is illustrated by a one-parameter family of Hurwitz-stable polynomials whose complex-conjugate zeros approach the imaginary axis.
ER -