Herglotz-Nevanlinna matrix functions and Hurwitz stability of matrix polynomials

TL;DR

This paper explores the connection between matrix-valued Herglotz-Nevanlinna functions and Hurwitz stable matrix polynomials, extending classical stability criteria with new theoretical insights and matrix extensions.

Contribution

It introduces a partial-fraction decomposition for self-adjoint rational matrix functions with Herglotz-Nevanlinna property and extends classical theorems to matrix cases.

Findings

Established a partial-fraction decomposition for matrix functions.

Connected Herglotz-Nevanlinna functions with Laurent series.

Extended classical theorems by Chebotarev and Grommer to matrices.

Abstract

This paper elaborates on a relationship between matrix-valued Herglotz-Nevanlinna functions and Hurwitz stable matrix polynomials, which generalizes the corresponding classical stability criterion. The main motivation comes from the author's recent stability studies linked with matricial Markov parameters. To fulfill our goals, we first give a partial-fraction decomposition of a self-adjoint rational matrix function with the Herglotz-Nevanlinna property. The next step is to connect a matrix-valued Herglotz-Nevanlinna function with its matricial Laurent series. Certain matrix extensions to two classical theorems by Chebotarev and Grommer, respectively, are also established.