# 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 Product

The Kronecker Product of a pair of matrices $A\in \mathbb {R} ^{m\times n}$  and $B\in \mathbb {R} ^{p\times q}$  is defined as follows:

$A\otimes B={\begin{bmatrix}a_{11}B&a_{12}B&\cdots &a_{1n}B\\a_{21}B&a_{22}B&\cdots &a_{2n}B\\\vdots &\vdots &\ddots &\vdots \\a_{m1}B&a_{m2}B&\cdots &a_{mn}B\end{bmatrix}}\in \mathbb {R} ^{mp\times nq}$ .

## Lemma 1: Manipulation Rules of Kronecker Product

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

• $1\otimes A=A$ ;
• $(A+B)\otimes C=A\otimes C+B\otimes C$
• $(A\otimes B)(C\otimes D)=(AC)\otimes (BD)$
• $(A\otimes B)^{T}=A^{T}\otimes B^{T}$
• $(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$
• $\lambda (A\otimes B)={\lambda _{i}(A)\lambda _{j}{B}}$

## Theorem

In terms of Kronecker products, the following theorem gives the $\mathbb {D}$ -stability condition for the general LMI region case: Let $\mathbb {D} =\mathbb {D} _{L,M}$  be an LMI region, whose characteristic function is

$F_{\mathbb {D} }=L+sM+{\overline {s}}M^{T}$

Then, a matrix $A\in \mathbb {R} ^{n\times n}$  is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix $P$  such that

$R_{\mathbb {D} }(A,P)=L\otimes P+M\otimes (AP)+M^{T}\otimes (AP)^{T}<0$ ,

where $\otimes$  represents the Kronecker product.

## Lemma 2

Given two LMI regions $\mathbb {D} _{1}$  and $\mathbb {D} _{2}$ , a matrix $A$  is both $\mathbb {D} _{1}$ -stable and $\mathbb {D} _{2}$ -stable if there exists a positive definite matrix $P$  , such that $R_{\mathbb {D} _{1}}(A,P)<0$  and $R_{\mathbb {D} _{2}}(A,P)<0$ . WIP, additional references to be added