Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x D A.t/x, where A is continuous on an interval Œa;1/. We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x D Pu, where P is continuously differentiable on Œa;1/ and u is a solution of a system u D B.t/u that has the property in question. In this casewe say that P preconditions the given system for the property in question. MSC: 15A99; 34C11; 34C41;