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Mat. Zametki, 1976, Volume 19, Issue 4, Pages 611–614 (Mi mz7780)  

Estimate for the spectrum of an operator bundle and its application to stability problems

V. I. Frolov

All-Union Scientific-Research Institute of Electric Power Engineering

Abstract: Simple estimates are obtained for the spectrum of the operator bundle $R(\lambda)=\sum_{i=0}^nA_{n-i}\lambda^i$ in terms of estimates of the maximum and minimum eigenvalues of the operators $\frac12(A_{n-i}+A_{n-i}^*)$ $(i=0,1,2,…,n)$ and the norms of the operators $\frac12(A_{n-i}-A_{n-i}^*)$ $(i=0,1,2,…,n)$. We formulate a criterion of the asymptotic stability of the differential equations
$$ \sum_{i=0}^nA_{n-i}\frac{d^{(i)}x}{dt^i}=0 $$
We present examples of the stability conditions for equations with $n=2$ and $n=3$.

Full text: PDF file (285 kB)

English version:
Mathematical Notes, 1976, 19:4, 369–371

Bibliographic databases:

UDC: 517.4
Received: 17.05.1973

Citation: V. I. Frolov, “Estimate for the spectrum of an operator bundle and its application to stability problems”, Mat. Zametki, 19:4 (1976), 611–614; Math. Notes, 19:4 (1976), 369–371

Citation in format AMSBIB
\Bibitem{Fro76}
\by V.~I.~Frolov
\paper Estimate for the spectrum of an operator bundle and its application to stability problems
\jour Mat. Zametki
\yr 1976
\vol 19
\issue 4
\pages 611--614
\mathnet{http://mi.mathnet.ru/mz7780}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=512575}
\zmath{https://zbmath.org/?q=an:0336.47001}
\transl
\jour Math. Notes
\yr 1976
\vol 19
\issue 4
\pages 369--371
\crossref{https://doi.org/10.1007/BF01156800}


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