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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1977, Volume 22, Issue 1, Pages 13–21 (Mi mz8020)

Generalized theorems of Liénard and Shepherd

G. F. Korsakov

Abstract: The paper considers a real polynomial $p(x)=a_0+a_1x+…+a_nx^n$ ($a_0>0$) for which there hold inequalities $\Delta_1>0, \Delta_3>0,…$ or $\Delta_2>0, \Delta_4>0$, where $\Delta_1,\Delta_2,…,\Delta_n$ are the Hurwitz determinants for polynomial $p(x)$. It is proven that polynomial $p(x)$ can have, in the right half-plane, only real roots, where the quantity of positive roots of polynomial $p(x)$ equals the quantity of changes of sign in the system of coefficients $a_0,a_2,…,a_n$, when $n$ is even, and $a_0,a_2,…,a_{n-1},a_n$, when $n$ is odd. From the proven theorem, in particular, there follows the Liénard and Shepherd criterion of stability.

Full text: PDF file (676 kB)

English version:
Mathematical Notes, 1977, 22:1, 498–503

Bibliographic databases:

UDC: 512

Citation: G. F. Korsakov, “Generalized theorems of Liénard and Shepherd”, Mat. Zametki, 22:1 (1977), 13–21; Math. Notes, 22:1 (1977), 498–503

Citation in format AMSBIB
\Bibitem{Kor77} \by G.~F.~Korsakov \paper Generalized theorems of Li\'enard and Shepherd \jour Mat. Zametki \yr 1977 \vol 22 \issue 1 \pages 13--21 \mathnet{http://mi.mathnet.ru/mz8020} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=476098} \zmath{https://zbmath.org/?q=an:0442.26009} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 1 \pages 498--503 \crossref{https://doi.org/10.1007/BF01147688}