Geodesic
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. 
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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/

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