# Lyapunov equation

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

$AXA^{H}-X+Q=0$ $AX+XA^{H}+Q=0$ .

The Lyapunov equation 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.

## Computational aspects of solution

Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used. For the continuous Lyapunov equation the method of Bartels and Stewart can be used.

## Analytic Solution

Defining the $\operatorname {vec} (A)$ operator as stacking the columns of a matrix $A$ and $A\otimes B$ as the Kronecker product of $A$ and $B$ , the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation. Furthermore, if the matrix $A$ is stable, the solution can also be expressed as an integral (continuous time case) or as an infinite sum (discrete time case).

### Discrete time

$(I-A\otimes A)\operatorname {vec} (X)=\operatorname {vec} (Q)$ $X=\sum _{k=0}^{\infty }A^{k}Q(A^{H})^{k}$ .

### Continuous time

Using again the Kronecker product notation and the vectorization operator, one has the matrix equation

$(I_{n}\otimes A+{\bar {A}}\otimes I_{n})\operatorname {vec} X=-\operatorname {vec} Q,$ $X=\int \limits _{0}^{\infty }e^{A\tau }Qe^{A^{H}\tau }d\tau$ .