# Almost-reducible linear system

of ordinary differential equations

A system

$$\tag{* } \dot{x} = A (t) x ,\ \ x \in \mathbf R ^ {n} ,$$

$$A ( \cdot ) : \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) ,$$

having the following property: There exist a system $\dot{y} = B y$, $y \in \mathbf R ^ {n}$, with constant coefficients and, for every $\epsilon > 0$, a Lyapunov transformation $L _ \epsilon (t)$ such that by the change of variables $x = L _ \epsilon (t) y$, the system (*) is transformed into the system

$$\dot{y} = ( B + C _ \epsilon (t) ) y ,$$

where

$$\sup _ {t \in \mathbf R } \ \| C _ \epsilon (t) \| < \epsilon .$$

Every reducible linear system is almost reducible.

#### References

 [1] N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 : 1 (1976) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146
How to Cite This Entry:
Almost-reducible linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-reducible_linear_system&oldid=45086
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article