On the geometric nature of partial and conditional stability
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 76-85
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We prove that certain problems which generalize the classical stability problem studied by A. M. Lyapunov admit a coordinate-free description. Namely, we mean problems on partial and conditional stability of solutions to vector functional differential equations, as well as a more general problem on the dependence of asymptotic properties of certain components of solutions on other ones.
For equations in the form
$$
x(t)-A\int^t_0x(s)d_sr(t,s)=f(t),
$$
where the matrix $A=\mathrm{const}$ and $r:\{(t,s):0\le s\le t\}\to\mathbb C$, the indicated types of stability are defined by properties of minimal subspaces of the vector space which are invariant with respect to a given transformation and belong to a given subspace.
Keywords:
functional differential equation, partial stability, conditional stability, invariant subspace.
@article{IVM_2008_3_a6,
author = {K. M. Chudinov},
title = {On the geometric nature of partial and conditional stability},
journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
pages = {76--85},
publisher = {mathdoc},
number = {3},
year = {2008},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/IVM_2008_3_a6/}
}
K. M. Chudinov. On the geometric nature of partial and conditional stability. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 76-85. http://geodesic.mathdoc.fr/item/IVM_2008_3_a6/