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Difference between revisions of "Lyapunov transformation"

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A non-degenerate linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061210/l0612101.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061210/l0612102.png" />), smoothly depending on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061210/l0612103.png" />, that satisfies the condition
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<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/l/l061/l061210/l0612104.png" /></td> </tr></table>
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A non-degenerate linear transformation  $  L ( t) : \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $(
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or  $  L ( t) : \mathbf C  ^ {n} \rightarrow \mathbf C  ^ {n} $),
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smoothly depending on a parameter  $  t \in \mathbf R $,
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that satisfies the condition
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$$
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\sup _ {t \in \mathbf R } [ \| L ( t) \| + \| L  ^ {-} 1 ( t)
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\| + \| \dot{L} ( t) \| ]  < + \infty .
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$$
  
 
It was introduced by A.M. Lyapunov in 1892 (see [[#References|[1]]]). The Lyapunov transformation is widely used in the theory of linear systems of ordinary differential equations. In many cases the requirement
 
It was introduced by A.M. Lyapunov in 1892 (see [[#References|[1]]]). The Lyapunov transformation is widely used in the theory of linear systems of ordinary differential equations. In many cases the requirement
  
<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/l/l061/l061210/l0612105.png" /></td> </tr></table>
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$$
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\sup _ {t \in \mathbf R }  \| \dot{L} ( t) \|  < + \infty
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$$
  
 
can be discarded.
 
can be discarded.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,  "Stability of motion" , Springer  (1967)  pp. 422</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,  "Stability of motion" , Springer  (1967)  pp. 422</TD></TR></table>

Revision as of 04:11, 6 June 2020


A non-degenerate linear transformation $ L ( t) : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $( or $ L ( t) : \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $), smoothly depending on a parameter $ t \in \mathbf R $, that satisfies the condition

$$ \sup _ {t \in \mathbf R } [ \| L ( t) \| + \| L ^ {-} 1 ( t) \| + \| \dot{L} ( t) \| ] < + \infty . $$

It was introduced by A.M. Lyapunov in 1892 (see [1]). The Lyapunov transformation is widely used in the theory of linear systems of ordinary differential equations. In many cases the requirement

$$ \sup _ {t \in \mathbf R } \| \dot{L} ( t) \| < + \infty $$

can be discarded.

References

[1] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)

Comments

References

[a1] W. Hahn, "Stability of motion" , Springer (1967) pp. 422
How to Cite This Entry:
Lyapunov transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_transformation&oldid=47733
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article