Lyapunov transformation

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