Lyapunov transformation

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

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

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

$$\underset{t\in \mathbf{R}}{sup}\Vert \dot{L}(t)\Vert <+\mathrm{\infty }$$

can be discarded.

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