$\{A_i,i\in\mathbb N\}$ is a fixed matrix sequence with element $A_i\in \mathbb R^{n\times m}$.
$\Phi\in\mathbb R^{m\times m}$ is a constant matrix and $d\in\mathbb R^m$ is a vector.
The sequence $\{g_i(d),i\in\mathbb N\}$ is given by the vector-valued function $g_i(d):\mathbb R^{m}\to \mathbb R^{n}$ with $g_0(d)={\bf 0}$ and \begin{align*} g_i(d)=\left\{ {\begin{array}{*{20}{l}} \max_{m\in\mathbb N_{[1,i]}}\{A_i\Phi^{i-m}d+g_{i-m}(d)\} ,\hspace{5em}&i\in \mathbb N_{[1,N-1]},\\ \max_{m\in\mathbb N_{[1,N]}}\{A_i\Phi^{i-m}d+g_{i-m}(d)\} ,&i\in \mathbb N_{\ge N}, \end{array}} \right. \end{align*} where $N\ge1$ is a positive integer.
The maximization of a vector-valued funtion is to be performed elementwise, e.g. $\max\{A_1d,A_2d\}=[\max\{A_1^{(1)}d,A_2^{(1)}d\},\cdots,\max\{A_1^{(n)}d,A_2^{(n)}d\}]^\top$, where $A_1^{(i)}$ and $A_2^{(i)}$ for $i\in \mathbb N_{[1,n]}$ are the $i^{\text{th}}$ rows of $A_1$ and $A_2$.
We can find that $g_i(d)$ is a continous function for $d$ because the function $\max\{\cdot\}$ is continous. If the limit $g_\infty(d)=\lim_{i\to\infty}g_i(d)$ is verified to be existent for some $d\in\mathbb R^m$, e.g. $d={\bf 1}=[1,1,\cdots,1]^\top$ and $g_\infty(\bf 1)$ exists, then $\textit{does the limit of } g_\infty(d) \textit{ exist for other } d\in\mathbb R^m$?
The limit of $g_\infty(d)$ implies the eigenvalues of $\Phi$ in the interior of the unit disc and the convergence of sequence $\{A_i,i\in\mathbb N\}$. Therefore, we can consider a more special problem further.
The sequence $\{g_i,i\in\mathbb N\}$ is given by the vector $g_i$ with $g_0={\bf 0}$ and \begin{align*} g_i=\left\{ {\begin{array}{*{20}{l}} \max_{m\in\mathbb N_{[1,i]}}\{\mathop \max \limits_{d\in R_m}A_0\Phi^{i-m}d+g_{i-m}\} ,\hspace{5em}&i\in \mathbb N_{[1,N-1]},\\ \max_{m\in\mathbb N_{[1,N]}}\{\mathop \max \limits_{d\in R_m} A_0\Phi^{i-m}d+g_{i-m}\} ,&i\in \mathbb N_{\ge N}, \end{array}} \right. \end{align*} where the optimization is to be performed elementwise, $A_0$ is fixed and bounded, $R_m$ is convex, compact set for $m\in \mathbb N_{[1,N]}$, $N\ge1$ is a positive integer and the eigenvalues of $\Phi$ in the interior of the unit disc. $\textit{Does the limit of }g_\infty=\lim_{i\to\infty}g_i\textit{ exist}$?