User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/alpha-Region via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, where

$$f_{D}(z):=\Lambda+z\Phi+\bar{z}\Phi^{T}= \{\lambda_{kl}+\phi_{kl}z+\phi_{lk}\bar{z}\}_{1\leq k,l\leq m}$$,

$$\Lambda\in\mathbb{S}^{m}$$, $$\Phi\in\mathbb{R}^{m \times m}$$, and $$\bar{z}$$ is the complex conjugate of $$z$$.

α-Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$\alpha\in\mathbb{R}_{>0}$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$, where $$\mathcal{H}(\alpha):=\{\lambda\in\mathbb{C}:Re(\lambda)<-\alpha\}$$ if and only if there exist $$X\in\mathbb{S}^{n}$$ and $$\epsilon\in\mathbb{R}_{>0}$$, where $$X>0$$, such that

$$AX+XA^{T}+2\alpha X<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$, $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -X & X \\ * & 0 & 0 \\ * & * & -\frac{1}{2}\alpha^{ -1}X \end{bmatrix} +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 \end{bmatrix} \end{Bmatrix}<0$$.

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}+2\alpha X<0$$

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -X & X \\ * & 0 & 0 \\ * & * & -\frac{1}{2}\alpha^{ -1}X \end{bmatrix} +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 \end{bmatrix} \end{Bmatrix}<0$$.