Lyapunov equation


The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems.
In particular, the discrete-time Lyapunov equation for is
where is a Hermitian matrix and is the conjugate transpose of, while the continuous-time Lyapunov equation is

Application to stability

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

The Lyapunov equation is linear; therefore, if contains entries, the equation can be solved in time using standard matrix factorization methods.
However, specialized algorithms are available which can yield solutions much quicker owing to the specific structure of the Lyapunov equation. For the discrete case, the Schur method of Kitagawa is often used. For the continuous Lyapunov equation the Bartels–Stewart algorithm can be used.

Analytic solution

Defining the vectorization 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 or as an infinite sum.

Discrete time

Using the result that, one has
where is a conformable identity matrix and is the element-wise complex conjugate of. 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
For comparison, consider the one-dimensional case, where this just says that the solution of is

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
which holds because
For comparison, consider the one-dimensional case, where this just says that the solution of is

Relationship Between Discrete and Continuous Lyapunov Equations

We start with the continuous-time linear dynamics:
And then discretize it as follows:
Where indicates a small forward displacement in time. Substituting the bottom equation into the top and shuffling terms around, we get a discrete-time equation for.
Where we've defined. Now we can use the discrete time Lyapunov equation for :
Plugging in our definition for, we get:
Expanding this expression out yields:
Recall that is a small displacement in time. Letting go to zero brings us closer and closer to having continuous dynamics—and in the limit we achieve them. It stands to reason that we should also recover the continuous-time Lyapunov equations in the limit as well. Dividing through by on both sides, and then letting we find that:
which is the continuous-time Lyapunov equation, as desired.