# Lyapunov equation

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

where is a Hermitian matrix and is the conjugate transpose of . The **continuous Lyapunov equation** is of form

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.

## Application to stability

In the following theorems , and and are symmetric. The notation means that the matrix is positive definite.

**Theorem** (continuous time version). Given any , there exists a unique satisfying if and only if the linear system is globally asymptotically stable. The quadratic function is a Lyapunov function that can be used to verify stability.

**Theorem** (discrete time version). Given any , there exists a unique satisfying if and only if the linear system is globally asymptotically stable. As before, is a Lyapunov function.

## Computational aspects of solution

Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used.^{[1]} For the continuous Lyapunov equation the method of Bartels and Stewart can be used.^{[2]}

## Analytic Solution

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

### Discrete time

Using the result that , one has

where is a conformable identity matrix.^{[3]} One may then solve for by inverting or solving the linear equations. To get , one must just reshape appropriately.

Moreover, if is stable, the solution can also be written as

### Continuous time

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

where denotes the matrix obtained by complex conjugating the entries of .

Similar to the discrete-time case, if is stable, the solution can also be written as