Let $A$, $B$, $K$ be $n\times n$, $n\times m$ and $m\times n$ matrix respectively. $\alpha_i>0$ is a scalar. Consider the following matrix:
$H_i=\int_0^\infty e^{(A+\alpha_i BK)^Tt}(\alpha_iI+\alpha_i^2K^TK)e^{(A+\alpha_i BK)t}dt$, where $A+\alpha_i BK$ is stable.
If $\alpha_i>\alpha_j$, is $H_i\geq H_j$? Namely, $H_i-H_j$ is positive semi-definite.
By numerical examples, it seems to be true, but is there a proof? Thanks in advance.