# Lyapunov equation

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

$A X A^{H} - X + Q = 0$

where $Q$ is a Hermitian matrix and $A^H$ is the conjugate transpose of $A$. The continuous Lyapunov equation is of form

$AX + XA^H + Q = 0$.

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.

## Application to stability

In the following theorems $A, P, Q \in \mathbb{R}^{n \times n}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite.

Theorem (continuous time version). Given any $Q>0$, there exists a unique $P>0$ satisfying $A^T P + P A + Q = 0$ if and only if the linear system $\dot{x}=A x$ is globally asymptotically stable. The quadratic function $V(z)=z^T P z$ is a Lyapunov function that can be used to verify stability.

Theorem (discrete time version). Given any $Q>0$, there exists a unique $P>0$ satisfying $A^T P A -P + Q = 0$ if and only if the linear system $x(t+1)=A x(t)$ is globally asymptotically stable. As before, $z^T P z$ 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 $\operatorname{vec} (A)$ 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

Using the result that $\operatorname{vec}(ABC)=(C^{T} \otimes A)\operatorname{vec}(B)$, one has

$(I-A \otimes A)\operatorname{vec}(X) = \operatorname{vec}(Q)$

where $I$ is a conformable identity matrix.[3] One may then solve for $\operatorname{vec}(X)$ by inverting or solving the linear equations. To get $X$, one must just reshape $\operatorname{vec} (X)$ appropriately.

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

$X = \sum_{k=0}^{\infty} A^{k} Q (A^{H})^k$.

### Continuous time

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

$(I_n \otimes A + \bar{A} \otimes I_n) \operatorname{vec}X = -\operatorname{vec}Q,$

where $\bar{A}$ denotes the matrix obtained by complex conjugating the entries of $A$.

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

$X = \int\limits_0^\infty e^{A \tau} Q e^{A^H \tau} d\tau$.