五次多项式在正Hadamard幂下的Hurwitz稳定性不变性
Invariance of Hurwitz-Stability of Polynomials of Degree Five under Positive Hadamard Powers
- Philadelphia University（费城大学）
- Jordan University of Science and Technology（约旦科技大学）
- University of Konstanz（康斯坦茨大学）
- HTWG Konstanz – University of Applied Sciences（康斯坦茨应用科学大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文完整刻画了五次Hurwitz稳定多项式分数Hadamard幂的保稳定性指数集，将其归结为单参数阈值p_*(f)，并证明该阈值光滑依赖且提供全局坐标。
AI中文摘要：
本文给出了五次首一Hurwitz稳定多项式f的分数Hadamard幂的保稳定性指数集的完整刻画。稳定性问题被归结为对单个标量函数的分析，该函数仅依赖于由多项式f的系数构成的三个参数。这一化简导出了一个唯一的稳定性阈值p_*(f)∈(0,1)，使得f的p次Hadamard幂是Hurwitz稳定的当且仅当p>p_*(f)。因此，保稳定性指数集恰好为(p_*(f),∞)。该阈值光滑依赖于参数，并在容许参数区域上提供了一个全局坐标。最后，通过一个单参数Hurwitz稳定多项式族（其共轭复零点趋近于虚轴）展示了光滑依赖性。
英文摘要：
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 $p_*(f) \in (0,1)$ 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 $(p_*(f),\infty)$. 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.
