User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Generalized Lyapunov Stability

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