On the stability of certain time-dependent dynamical systems (Q1076215)

The author is concerned with the asymptotic stability of the nonautonomous differential system (1) \(X'=A(t)X\), where \(t\to A(t)\) is an \(n\times n\) real continuous function matrix, as well as with the generalization of asymptotic stability from (1) to the perturbed system \(X'(t)=A(t)X(t)+f(t,x).\) According to a classical result, the asymptotic stability of (1) can be characterized in terms of its fundamental matrix F(t). Unfortunately, F(t) cannot be effectively determined, except for some concrete cases (most of them for \(n=2)\). The main advantage of this paper over the classical results consists in giving criteria of asymptotic stability of (1) directly in terms of A(t). The author gives examples of systems whose asymptotic stability can be proved via his criteria (but not by classical method - reviewer's remark). Therefore, the results of this paper have to be known by everybody working in stability of differential systems. In the autonomous case \(A(t)=A\), the characterization of componentwise asymptotic stability in terms of the elements \(a_{ij}\) of A, is given by Voicu and the reviewer [An. Ştiinţ. Univ. Al. I. Cuza Iasi, N. Ser. Sect. Ia (to appear)].