On the relation between Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions. (English) Zbl 1524.30015
Summary: In this paper, we elaborate on the relationship between the Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions. Our strategy is that, for a monic matrix polynomial, we associate a rational matrix-valued function with its even-odd split and then check the Hurwitz stability of the matrix polynomial by testing the Stieltjes property of the related rational matrix-valued function. On the basis of this relationship, we present matrix generalizations of a classical stability criterion by Gantmacher, Chebotarev theorem, Grommer theorem and some aspects of the modified Hermite-Biehler theorem. Our work is motivated by one of the authors’ recent stability studies linked with matricial Markov parameters.
MSC:
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 15A54 | Matrices over function rings in one or more variables |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 65F60 | Numerical computation of matrix exponential and similar matrix functions |
| 65F99 | Numerical linear algebra |
| 93D20 | Asymptotic stability in control theory |
Keywords:
matrix polynomials; Hurwitz stability; Herglotz-Nevanlinna functions; Stieltjes functions; Hankel matrices; Hermite-Biehler theoremSoftware:
Polynomial ToolboxReferences:
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