Hadamard Factorization of Hurwitz Stable Polynomials

The Hurwitz stable polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p, q ∈ R[x]: p(x) = anx n + an−1x n−1 + · · ·+ a1x + a0 q(x) = bmx m + bm−1x m−1 + · · ·+ b1x + b0 the Hadamard product (p ∗ q) is defined as (p ∗ q)(x) = akbkx + ak−1bk−1x + · · ·+ a1b1x + a0b0 where k = min(m,n). Some results (see [16]) shows that if p, q ∈ R[x] are stable polynomials then (p ∗ q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n ≥ 4 (see [15]). In this work we will give some conditions to Hadamard factorization existence for stable polynomials.