To Show This Is Positive Semidefinite

Given that $A\in M_n(\mathbb{R})$ is asymptotically stable and we define $P=\int_{0}^{\infty} e^{A^Tt}Qe^{At}dt$. I need to show that $P$ is positive semidefinite if $Q$ is positive semidefinite. $Q$ is symmetric too.

$Q$ is psd thus $x^TQx\ge 0\Rightarrow x^Te^{A^Tt}Qe^{At}x\ge0\Rightarrow\int_{0}^{\infty}x^Te^{A^Tt}Qe^{At}x dt\ge 0$ Is this the way to show that? Thanks.

Also, I we define: $P= \sum_{0}^{\infty}(A^T)^iQ A^i$, I need to show $P$ is psd if $Q$ is psd. please help.

• Nice Question. I am not sure about the part where you bring $x$ inside the integral, otherwise it looks fine. Is this from an application? – dineshdileep May 12 '17 at 6:46