Let $A \in R^{n \times n}$ and $B \in R^{n \times m}$. Define
$$Q_t = \int_{0}^t e^{sA}BB^T e^{sA^T} ds$$
Suppose that $x \in \text{Im } Q_t$, ie, $\exists \eta \in R^n$ such that $$x = Q_t \eta$$
Show that if $t_1 \geq t$, then $x$ is also on $\text{Im } Q_{t_1}$, ie,
$$\text{Im } Q_{t} \subseteq \text{Im } Q_{t_1}$$
My attempts can be seen here. I didn't make much progress after that.
Thanks in advance!
Let $M(s) = e^{sA}BB^Te^{sA^T}$. Note that $M(s)$ is (symmetric) positive semidefinite. It follows that $Q_t = \int_0^tM(s)\,ds$ is positive semidefinite.
Noting that $\operatorname{Im}Q_t = \ker(Q_t)^\perp$, it suffices to show that $\ker Q_t \supseteq \ker Q_{t_1}$. In order to do that, it suffices to note that $$ Q_t x = 0 \iff\\ x^TQ_tx = 0 \iff\\ \int_0^t x^T M(s)x\,ds = 0 $$ Note, however, that $f(s) = x^TM(s)x$ is a non-negative function.