## 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.
Reviewer: E.Barvínek (Brno)

### MSC:

 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.

### Keywords:

asymptotic diagonalization; linear system