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Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials
Mathematics of Control, Signals, and Systems ( IF 0.976 ) Pub Date : 2019-09-05 , DOI: 10.1007/s00498-019-00245-8
Baltazar Aguirre-Hernández, Martín Eduardo Frías-Armenta, Jesús Muciño-Raymundo

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic \(\mathbb {C}\)-actions \(\mathscr {A}\) on the space of polynomials of degree n. For each orbit \(\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}\) of \(\mathscr {A}\), we study the dynamical problem of the existence of a complex rational vector field \(\mathbb {X}(z)\) on \(\mathbb {C}\) such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit \(\{ s \cdot P(z) = 0 \}\). Regarding the above \(\mathbb {C}\)-action coming from the \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure, we prove the existence of a complex rational vector field \(\mathbb {X}(z)\) on \(\mathbb {C}\), which describes the geometric change of the n-root configuration in the unitary disk \(\mathbb {D}\) of a \(\mathbb {C}\)-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in \(\mathbb {C}\backslash \overline{\mathbb {D}}\), by constructing a principal \(\mathbb {C}^* \times \mathbb {S}^1\)-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.
更新日期:2019-09-05

 

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