New Versions of the Hermite Bieler Theorem in Stability Contexts
American Journal of Applied Mathematics
Volume 7, Issue 1, February 2019, Pages: 1-4
Received: Jan. 3, 2019; Accepted: Jan. 24, 2019; Published: Feb. 25, 2019
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Author
Ziad Zahreddine, Mathematics Division, College of Science & Information Systems, Rafik Hariri University, Mechref, Damour, Lebanon
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Abstract
The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.
Keywords
Hermite-Bieler Theorem, Routh-Hurwitz Criterion, Schur-Cohn Criterion, Stability Analysis, Positive Para-Odd Functions, Conformal Mappings
To cite this article
Ziad Zahreddine, New Versions of the Hermite Bieler Theorem in Stability Contexts, American Journal of Applied Mathematics. Vol. 7, No. 1, 2019, pp. 1-4. doi: 10.11648/j.ajam.20190701.11
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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