Asymptotic diagonalization of a linear ordinary differential system. (English) Zbl 0806.34009

The authors deal with the asymptotic diagonalization of the linear system \(dy/dt = A(t)y\), \(t \to \infty\), \(y \in \mathbb{R}^ n\), where \(A(t) = \Lambda (t) + R(t)\) is an \(n \times n\) continuous matrix with \(\Lambda (\cdot)\) a diagonal continuous matrix. Under assumptions which are partially weaker and partially stronger than assumptions used by other authors, they obtain the diagonalized system \(dz/dt = (\Lambda (t) + \text{diag} R(t))z\) by means of the transformation \(y = (I_ n + Q(t))z\) where \(Q(\cdot)\) is a convenient \(n \times n\) smooth matrix such that \(\lim_{t \to \infty} Q(t) = 0\). The usefulness of the theorem is exhibited on four examples.


34A30 Linear ordinary differential equations and systems
34D05 Asymptotic properties of solutions to ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.