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Difference between revisions of "Reducible linear system"

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''of ordinary differential equations''
 
''of ordinary differential equations''
  
 
A system
 
A system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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\dot{x}  = A ( t) x,\ \
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x \in \mathbf R  ^ {n} \  ( \textrm{ or }  \mathbf C  ^ {n} ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804002.png" /></td> </tr></table>
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$$
 +
A ( \cdot ): \mathbf R  \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n}
 +
) \  ( \textrm{ or }  \mathop{\rm Hom} ( \mathbf C  ^ {n} , \mathbf C  ^ {n} )),
 +
$$
  
that can be transformed into a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804003.png" /> with constant coefficients by a change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804004.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804005.png" /> is a [[Lyapunov transformation|Lyapunov transformation]]. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804006.png" /> is continuous and periodically depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804007.png" />, then (*) is a reducible system (Lyapunov's theorem). The system (*) is reducible if and only if there is a Lyapunov transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804008.png" /> and an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r0804009.png" /> such that every solution of (*) has the form
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080400/r08040010.png" /></td> </tr></table>
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$$
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x ( t)  = L ( t) e ^ {tB } x ( 0)
 +
$$
  
 
(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
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article