On the class of proper linear differential systems with unbounded coefficients

Proper linear differential systems (whose coefficients are not necessarily bounded on the half-line) are defined as systems for which there exists a generalized Lyapunov transformation reducing them to a diagonal system with constant coefficients (Basov). We prove that Lyapunov’s original definition of a proper system and the Perron and Vinograd criteria hold for the class of proper systems as well as for the class of proper systems with uniformly bounded coefficients. We show that the Lyapunov properness criterion for a triangular system fails for systems with unbounded coefficients; namely, we construct an improper system with the following properties: the Lyapunov exponents of all nonzero solutions of that system are finite and exact, and for an arbitrary reduction of this system by a generalized Lyapunov transformation to triangular form, its diagonal coefficients have finite exact mean values, whose set with regard of multiplicities is independent of the choice of the transformation. In addition, we show that the main property of proper systems with uniformly bounded coefficients (preservation of conditional exponential stability as well as the dimension of the exponentially stable manifold and the exponent of the asymptotic behavior of solutions under perturbations of higher-order smallness) holds for proper systems with unbounded coefficients as well.