|
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 $\frac12(A_{n-i}+A_{n-i}^*)$ $(i=0,1,2,\dots,n)$ and the norms of the operators $\frac12(A_{n-i}-A_{n-i}^*)$ $(i=0,1,2,\dots,n)$. 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 $n=3$.
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
Linking options:
https://www.mathnet.ru/eng/mzm7780 https://www.mathnet.ru/eng/mzm/v19/i4/p611
|
|