Is it un/common to write the state of a dynamical system in the following manner:
$$ \begin{pmatrix}x_{t+1} \\ v_{t+1} \end{pmatrix} = \begin{pmatrix}A & B \\ C & D \end{pmatrix}\begin{pmatrix}x_{t} \\ v_{t} \end{pmatrix} + \begin{pmatrix}E & F \\ G & H \end{pmatrix}\begin{pmatrix}p_{t} \\ q_{t} \end{pmatrix}$$
What could this form mean? And how could this impact the way we analyze the system? (For example, if we want to check stability, etc)
Your insights will help a lot.
Note that $p_t$ and $q_t$ are not states. So this equation is just
$$z_{t+1} = M z_t + N u_t$$
where $z_t := \begin{pmatrix}x_{t+1} \\ v_{t+1} \end{pmatrix}$, $u_t := \begin{pmatrix}p_{t} \\ q_{t} \end{pmatrix}$, $M := \begin{pmatrix}A & B \\ C & D \end{pmatrix}$ and $N := \begin{pmatrix}E & F \\ G & H \end{pmatrix}$. So the stability can be determined by checking the eigenvalues of $M$.