LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation-Based Properties

Young’s Relation-Based Properties

1. Consider ${\displaystyle X}$ ,${\displaystyle Y\in \mathbb {R} ^{n\times m}}$  and ${\displaystyle Z\in \mathbb {S} ^{m}}$ . The matrix inequality given by

{\displaystyle {\begin{aligned}Z+XTY+YTX>0,\end{aligned}}}
is satisfied if and only if there exist ${\displaystyle Q\in \mathbb {S} ^{m}}$ , ${\displaystyle P\in \mathbb {S} ^{n}}$ ,${\displaystyle G1\in \mathbb {R} ^{n\times n}}$ ,${\displaystyle G2\in \mathbb {R} ^{n\times m}}$ ,${\displaystyle F\in \mathbb {R} ^{m\times n}}$ , and ${\displaystyle H\in \mathbb {R} ^{m\times m}}$ , where ${\displaystyle Q>0}$
and ${\displaystyle P>0}$ , such that
${\displaystyle {\begin{bmatrix}P&Y\\*&Q\end{bmatrix}}>0}$  and ${\displaystyle {\begin{bmatrix}Z+Q+X^{T}PX&F-X^{T}G_{2}&H-X^{T}G_{1}\\*&G_{1}+G_{1}^{T}-P&F^{T}+G_{2}-Y\\*&*&H^{T}+H-Q\end{bmatrix}}}$

2. Consider ${\displaystyle X}$ ,${\displaystyle Y\in \mathbb {R} ^{n\times n}}$  and ${\displaystyle W\in \mathbb {S} ^{m}}$ , where ${\displaystyle X}$  is full rank and ${\displaystyle W>0}$ . The matrix inequality given by

{\displaystyle {\begin{aligned}X^{T}-W>0,\end{aligned}}}
is satisfied if there exist ${\displaystyle \lambda \in \mathbb {R} _{>0}}$  such that
${\displaystyle {\begin{bmatrix}\lambda \mathbf {1} &\lambda \mathbf {1} &\mathbf {0} \\*&\mathbf {X} +\mathbf {X} ^{T}&\mathbf {W} ^{\frac {1}{2}}\\*&*&\lambda \mathbf {1} \end{bmatrix}}>0}$