LMIs in Control/Click here to continue/LMIs in system and stability Theory/General LMI Region D-Admissibility

The System
Consider $$A, E \in \mathbb{R}^{n \times n}$$. The pair ($$E$$,$$A$$) is D-admissible if it is regular and causal, and the eigenvalues of ($$E$$,$$A$$) lie within the LMI region D of the complex plane, which is defined as

$$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 complex conjugate of $$z$$.

Conditions
The pair ($$E$$,$$A$$) is D-admissible if and only if any of the following equivalent conditions are satisfied.


 * 1) There exist $$P \in \mathbb S^n, S \in \mathbb R^{(n-n_e)\times (n-n_e)},$$ $$U, V \in \mathbb R^{n \times (n-n_e)},$$ where $$n_e = $$  rank$$(E), \mathcal R(U)=\mathcal N(E^T), \mathcal R(V)=\mathcal N(E),$$and $$P>0,$$ satisfying   $$[\lambda_{kl}EPE^T+\phi_{kl}E^TPA^T+AVSU^T+US^TV^TA^T]_{1 \leq k,l \leq m}<0,$$
 * 2) There exist $$P,Q \in \mathbb S^n,$$ where $$P>0,$$ satisfying $$E^TQE \geq 0$$ and  $$[\lambda_{kl}EPE^T+\phi_{kl}APE+\phi_{lk}E^TPA^T+A^TQA]_{1\leq k,l\leq m}<0,$$
 * 3) There exist $$P \in \mathbb S^n, S \in \mathbb R^{(n-n_e)\times (n-n_e)},U \in \mathbb R^{n \times (n-n_e)},$$ where $$n_e=$$ rank$$(E),UE=0,$$ and $$P>0,$$ satisfying $$[\lambda_{kl}EPE^T+\phi_{kl}APE+\phi_{lk}E^TPA^T+A^TU^TSUA]_{1\leq k,l\leq m}<0,$$
 * 4) There exist $$P \in \mathbb S^n, S \in \mathbb R^{(n-n_e)\times (n-n_e)},$$$$U, V \in \mathbb R^{n \times (n-n_e)},$$ where $$n_e=$$  rank$$(E), \mathcal R(U)=\mathcal N(E^T), \mathcal R(V)=\mathcal N(E),$$ and $$P>0,$$ satisfying   $$\Lambda \otimes EPE^T+\Phi \otimes (APE)+\Phi^T\otimes (EPA^T)+1_{mm} \otimes (AVSU^T+US^TV^TA^T)<0,$$ where $$\otimes $$ is the Kroenecker product and $$1_{mm}$$ is an $$m \times m$$ matrix filled with ones.
 * 5) There exist $$P,Q \in \mathbb S^n,$$ where $$P>0,$$ satisfying $$E^TQE \geq 0$$ and $$\Lambda \otimes EPE^T+\Phi \otimes (APE)+\Phi^T\otimes (EPA^T)+1_{mm} \otimes (A^TQA)<0,$$ where $$\otimes $$ is the Kroenecker product and $$1_{mm}$$ is an $$m \times m$$ matrix filled with ones.
 * 6) There exist $$P \in \mathbb S^n, S \in \mathbb R^{(n-n_e)\times (n-n_e)},U \in \mathbb R^{n \times (n-n_e)},$$ where $$n_e=$$ rank$$(E),UE=0,$$ and $$P>0,$$ satisfying $$\Lambda \otimes EPE^T+\Phi \otimes (APE)+\Phi^T\otimes (EPA^T)+1_{mm} \otimes (A^TU^TSUA)<0,$$ where $$\otimes $$ is the Kroenecker product and $$1_{mm}$$ is an $$m \times m$$ matrix filled with ones.

Reference
Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.