Reducibility of linear systems with aftereffect

It is shown that a linear system with aftereffect on each finite-dimensional subspace of solutions with finite Lyapunov indices is asymptotically similar under natural assumptions to a system of ordinary differential equations. A system with the right-hand side recurrent with respect to time is investigated in detail and a family of systems with aftereffect, whose space of solutions with finite Lyapunov indices is finite-dimensional, is constructed. The research is based on the conception of N. N. Krasovskii, according to which to every system with aftereffect there corresponds some dynamical system with infinite-dimensional phase space and a flow on it generated by solutions of the original system with aftereffect.