The geometric nature of partial and conditional stability (Q1006950)

From the author's abstract: 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 \[ \dot x(t)-A\int_{0}^{t}x(s)\,d_{s}r(t,s)=f(t), \] where \(A=\text{const}\) and \(r:\{(t,s): 0\leq s\leq t\}\to {\mathcal C}\), 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.