Difference between revisions of "Almost-reducible linear system"
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''of ordinary differential equations'' | ''of ordinary differential equations'' | ||
A system | 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 | + | 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|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 | where | ||
| − | < | + | $$ |
| + | \sup _ {t \in \mathbf R } \ | ||
| + | \| C _ \epsilon (t) \| | ||
| + | < \epsilon . | ||
| + | $$ | ||
Every [[Reducible linear system|reducible linear system]] is almost reducible. | Every [[Reducible linear system|reducible linear system]] is almost reducible. | ||
Latest revision as of 16:10, 1 April 2020
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
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