# LMIs in Control/pages/Generalized Lyapunov Theorem

*WIP, Description in progress*

The theorem can be viewed as a true essential generalization of the well-known continuous- and discrete-time Lyapunov theorems.

## Kronecker ProductEdit

The Kronecker Product of a pair of matrices and is defined as follows:

.

## Lemma 1: Manipulation Rules of Kronecker ProductEdit

Let be matrices with appropriate dimensions. Then, the Kronecker product has the following properties:

- ;

## TheoremEdit

In terms of Kronecker products, the following theorem gives the -stability condition for the general LMI region case: Let be an LMI region, whose characteristic function is

Then, a matrix is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix such that

,

where represents the Kronecker product.

## Lemma 2Edit

Given two LMI regions and , a matrix is both -stable and -stable if there exists a positive definite matrix , such that and .

*WIP, additional references to be added*

## External LinksEdit

A list of references documenting and validating the LMI.