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On stable polynomials. (English) Zbl 0703.12001
This is an expository paper about the Hurwitz-Routh criterion on the Hurwitz (or stable) polynomials [A. Hurwitz, Math. Ann. 46, 273-284 (1895; JFM 26.0119.03)]. It contains a detailed discussion of the works of Routh, Hurwitz, Schur, Pontryagin and others. There are occasionally obtained new proofs of classical results.
Reviewer: D.Ştefănescu
MSC:
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
26C10 Real polynomials: location of zeros
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:
[1] Ch. Hermite: Sur le nombre des racines d’une équation algébrique comprises entre des limites données. Crelles J. 52, 39 (1856).
[2] J. Routh: A treatise on the stability of a given state of motion. London 1877.
[3] A. Hurwitz: Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt. Math. Ann. 46, 273 (1895). · Zbl 0962.01500 · doi:10.1007/BF01446812
[4] J. Schur: Über die algebraischen Gleichungen, die nur Wurzeln mit negativen Realteilen besitzen. Z. angew. Math. Mech, 1, 307 (1921). · JFM 48.0082.03 · doi:10.1002/zamm.19210010405
[5] L. S. Pontryagin: On the zeros of some elementary transcendental functions. (Russian) Izv. Ak. Nauk SSSR, Ser. Mat. 6 (1942), 115-134. · Zbl 0068.05803
[6] H. Cremer F. H. Effertz: Über die algebraischen Kriterien für die Stabilität von Regulungsystemen. Math. Ann. 137 (1959), 328-350. · Zbl 0092.01604 · doi:10.1007/BF01360969 · eudml:160686
[7] R. Bellman: Introduction to Matrix Analysis. Mc Graw-Hill Book Company, New York 1960. · Zbl 0124.01001
[8] F. R. Gantmacher: Theory of Matrices. (Russian) Izd. Nauka, Moskva 1966. · Zbl 0136.00410
[9] B. P. Demidowich: Lectures on the Mathematical Theory of Stability. (Russian) Izd. Nauka, Moskva 1967.
[10] J. Hale: Theory of Functional Differential Equations. Springer-Verlag, New York 1977. · Zbl 0352.34001
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