On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials
Abstract
A real scalar polynomial whose roots lie in the left half--plane of the complex plane is called a Hurwitz polynomial. This notion goes back to the works of J. C. Maxwell, E. J. Routh, and A. Hurwitz, and was later studied using techniques such as Sturm sequences, Markov parameters, and continued fractions; see, for instance, \emph{The Theory of Matrices}, Vol. II, Chapter XV, AMS Chelsea (2000), by F. Gantmacher. A $q\times q$ matrix polynomial $P(z)$ is called Hurwitz if $\det P(z)$ is a Hurwitz polynomial. Every matrix polynomial $f_n$ can be written in the form $f_n(z)=h_n(z^2)+z\,g_n(z^2)$. The matrix polynomial $f_{2m}$ is said to be of Hurwitz type if the expression $g_{2m}(z)h_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $f_{2m+1}$ is of Hurwitz type if $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$ has the same property. The concept of Hurwitz-type matrix polynomials was introduced in \emph{On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials}, Linear Algebra Appl. 476 (2015), by A. E. Choque Rivero.
In the present work, we derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. The fact that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials was suggested and partially proved using Bezoutians in \emph{On generalization of classical Hurwitz stability criteria for matrix polynomials}, J. Comput. Appl. Math. 383 (2021), by X. Zhan and A. Dyachenko.
In contrast to that work, we employ the decomposition of the Bezoutian form introduced in \emph{Some Questions in the Theory of Moments}, Translations of Mathematical Monographs~2, AMS, 1962, by N. I. Akhiezer and M. G. Krein, for scalar polynomials.
Additionally, we propose a method to enlarge the class of Hurwitz-type matrix polynomials by adding to a given polynomial a matrix polynomial that is not of Hurwitz type, so that the resulting polynomial becomes of Hurwitz type.
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References
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