Difference between revisions of "Reducible linear system"
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''of ordinary differential equations'' | ''of ordinary differential equations'' | ||
A system | A system | ||
| − | + | $$ \tag{* } | |
| + | \dot{x} = A ( t) x,\ \ | ||
| + | x \in \mathbf R ^ {n} \ ( \textrm{ or } \mathbf C ^ {n} ), | ||
| + | $$ | ||
| − | + | $$ | |
| + | A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} | ||
| + | ) \ ( \textrm{ or } \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} )), | ||
| + | $$ | ||
| − | that can be transformed into a system | + | that can be transformed into a system $ \dot{y} = By $ |
| + | with constant coefficients by a change of variables $ x = L ( t) y $, | ||
| + | where $ L ( t) $ | ||
| + | is a [[Lyapunov transformation|Lyapunov transformation]]. If the mapping $ A ( t) $ | ||
| + | is continuous and periodically depends on $ t $, | ||
| + | then (*) is a reducible system (Lyapunov's theorem). The system (*) is reducible if and only if there is a Lyapunov transformation $ L ( t) $ | ||
| + | and an operator $ B $ | ||
| + | such that every solution of (*) has the form | ||
| − | + | $$ | |
| + | x ( t) = L ( t) e ^ {tB } x ( 0) | ||
| + | $$ | ||
(Erugin's criterion). | (Erugin's criterion). | ||
Latest revision as of 08:10, 6 June 2020
of ordinary differential equations
A system
$$ \tag{* } \dot{x} = A ( t) x,\ \ x \in \mathbf R ^ {n} \ ( \textrm{ or } \mathbf C ^ {n} ), $$
$$ A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) \ ( \textrm{ or } \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} )), $$
that can be transformed into a system $ \dot{y} = By $ with constant coefficients by a change of variables $ x = L ( t) y $, where $ L ( t) $ is a Lyapunov transformation. If the mapping $ A ( t) $ is continuous and periodically depends on $ t $, then (*) is a reducible system (Lyapunov's theorem). The system (*) is reducible if and only if there is a Lyapunov transformation $ L ( t) $ and an operator $ B $ such that every solution of (*) has the form
$$ x ( t) = L ( t) e ^ {tB } x ( 0) $$
(Erugin's criterion).
References
| [1] | A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian) |
| [2] | N.P. Erugin, "Reducible systems" Trudy Mat. Inst. Steklov. , 13 (1946) (In Russian) |
How to Cite This Entry:
Reducible linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reducible_linear_system&oldid=18924
Reducible linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reducible_linear_system&oldid=18924
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article