Lyapunov reducibility of infinitesimal perturbations of equations and systems

We consider two classes of infinitesimally small perturbations of a given linear differential equation with continuous, possibly unbounded, coefficients. The first class consists of its perturbations in the space of all linear systems and the second class consists of perturbations with somewhat slower decay but in a narrower space, namely the space of systems corresponding to single equations. It is shown that the values of a Lyapunov invariant functional on the first class belong to the range of the same functional on the second class. For systems with bounded coefficients, it is shown that the said sets coincide.