Lyapunov equation

From Freepedia

In control theory, the discrete Lyapunov equation is a system of the form

<math>A X A^H - X + Q = 0</math>

where <math>Q</math> is a hermitian matrix. It occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov.

The discrete Lyapunov equations can, by using Schur complements, be written as

<math>\begin{bmatrix}

X^{-1} & A \\ A^H & X-Q \end{bmatrix}=0</math> or equivalently as

<math>\begin{bmatrix}

X & XA \\ A^HX & X-Q \end{bmatrix}=0</math>.

The continuous Lyapunov equations are of form

<math>AX + XA^H + Q = 0</math>.