Solving Riccati-like matrix inequality How can I find P2 in the following inequality?
$$S_{ai} = Q + G P_2 G'- P_2 - \bar{\gamma}^2 G P_2 C' R^{-1} C P_2 G' \prec 0$$
where $R = \beta C P_2 C'+ V$, $\beta=0.9$, $\bar{\gamma} = 0.9$ and
G =
0.5437    0.0768
0.2040   -1.1470

C =
1     0
0     1

Q =
0.1363    0.1527
0.1527    2.9000

Please help me. I think it is not solvable by LMI. please help me if there is any solution for solving this inequality in your idea.

$$
\begin{align}
Q + G P_2 G'- P_2 - \bar{\gamma}^2 G P_2 C' R^{-1} C P_2 G' &\prec 0 \\[5mm]&\Updownarrow \textrm{if }R\prec 0\\[5mm]
\begin{pmatrix}Q + G P_2 G'- P_2 &\bar{\gamma} G P_2 \\ \bar{\gamma}P_2 G'& \beta(C'C)P_2(C'C)\end{pmatrix}&\prec 0 
\end{align}
$$
