A Perron-like stability criterion for linear systems

Abstract

The purpose of this note is to give the solution of a certain stability problem stated by J. Kloch. It is shown that the zero solution of the homogeneous linear system ~ = A(t)x (x e ~n t e ~+ = [0, + ~)) is stable (in the Liapunov sense) provided that those solutions of the non-homogeneous system ~=A(t)x+ u (u e R n an arbitrary constant) which correspond to vanishing initial perturbations are bounded. But stability is not included when only the initial time to = 0 is taken into account. Let be given the homogeneous linear vector differential equation