# Lyapunov transformation

From Encyclopedia of Mathematics

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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 [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) |

[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=53916

This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article