Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two $n\times m$ real matrices of full column rank (i.e. $\mathrm{rank}\, B_1=\mathrm{rank}\, B_2=m$) and define $$ \mathcal{R}_1 :=\left[B_1\, |\, A_1B_1\, |\, A_1^2B_1\, |\, \cdots\, |\, A_1^{n-1}B_1\right] $$ $$ \mathcal{R}_2 :=\left[B_2\, |\, A_2B_2\, |\, A_2^2B_2\, |\, \cdots\, |\, A_2^{n-1}B_2\right] $$ (In control theory the above-defined matrices are called reachability or controllability matrices of the pairs $(A_1,B_1)$ and $(A_2,B_2)$, resp.)
Question. Under the assumption that $\mathrm{rank}\,\mathcal{R}_1=\mathrm{rank}\,\mathcal{R}_2=n$, can we conclude that the following series $$ \sum_{k=0}^\infty A_1^k\, B_1\, B_2^\top\, (A_2^\top)^k $$ is non-singular? If not, do there exist some non-trivial conditions on the pairs $(A_1,B_1)$ and $(A_2,B_2)$ that guarantee that the above series is non-singular?
Remark. In case $A:=A_1=A_2$ and $B:=B_1=B_2$, then the above series coincides with the (unique) solution of the following discrete-time Lyapunov equation $$ X-A X A^\top = BB^\top. $$ It is well-know that, in this case, the condition $\mathrm{rank}\,\mathcal{R_1}=n$ implies that $X$ is positive definite (and, therefore, non-singular).
