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 $p_{\ast }(f)\in (0,1)$ such that the $p$-th Hadamard power of $f$ is Hurwitz stable if and only if $p>p_{\ast }(f)$. Consequently, the stability-preserving exponent set is precisely $(p_{\ast }(f),\infty )$. 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.

9 pages, one figure