Difference between revisions of "Lyapunov transformation"
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| + | 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 [[#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 | ||
| − | < | + | $$ |
| + | \sup _ {t \in \mathbf R } \| \dot{L} ( t) \| < + \infty | ||
| + | $$ | ||
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> | ||
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====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
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