Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

Baltazar Aguirre-Hernández, Martín Eduardo Frías-Armenta, Jesús Muciño-Raymundo

Research output: Contribution to journalArticlepeer-review

Abstract

© 2019, Springer-Verlag London Ltd., part of Springer Nature. We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal C× S1-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic C-actions A on the space of polynomials of degree n. For each orbit {s·P(z)|s∈C} of A, we study the dynamical problem of the existence of a complex rational vector field X(z) on C such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit { s· P(z) = 0 }. Regarding the above C-action coming from the C× S1-bundle structure, we prove the existence of a complex rational vector field X(z) on C, which describes the geometric change of the n-root configuration in the unitary disk D of a C-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in C\ D¯ , by constructing a principal C∗× S1-bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.
Original languageAmerican English
Pages (from-to)545-587
Number of pages43
JournalMathematics of Control, Signals, and Systems
DOIs
StatePublished - 1 Dec 2019

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