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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.

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.
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