Asymptotic stability of non-autonomous upper triangular systems and a generalization of Levinson s theorem

This article studies asymptotic stability of non-auto nomous linear systems with time-dependent coefficient matrices {A(t)}t∈R. The classical theorem of Levinson has been widely used to science and engineering non-autonomous systems, but systems with defective eigenvalues could not be covered because such a family does not allow continuous diagonalization. We study systems where the family allows to have upper triangulation and to have defective eigenvalues. In addition to the wider applicability, working with upper triangular matrices in place of Jordan form matrices offers more flexibility. We interpret our and earlier works including Levinson s theorem from the perspective of invariant manifold theory.