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. 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

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