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### Modification
The optimal value functions $- x' P_i x - \rho_i$ for $i = 1, \ldots, n$ satisfy the $N$ interrelated Bellman equations
\begin{align*}
-x' P_i x - \rho_i & = \max_u - \biggl[ x'R_i x + u' Q_i u + 2 u' W_i x \cr &
\beta \sum_j \Pi_{ij}E ((A_i x + B_i u + C_i w)' P_j (A_i x + B_i u + C_i w) x + \rho_j) \biggr]
\end{align*}
-> \begin{align*}
-x' P_i x - \rho_i & = \max_u - \biggl[ x'R_i x + u' Q_i u + 2 u' W_i x \cr &
-\beta \sum_j \Pi_{ij}E ((A_i x + B_i u + C_i w)' P_j (A_i x + B_i u + C_i w) x + \rho_j) \biggr]
\end{align*}
Not sure whether it should be a + or a -
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Modification
After the sentence$- x' P_i x - \rho_i$ for $i = 1, \ldots, n$ satisfy the $N$ interrelated Bellman equations
The optimal value functions
\begin{align*}
-x' P_i x - \rho_i & = \max_u - \biggl[ x'R_i x + u' Q_i u + 2 u' W_i x \cr &
\beta \sum_j \Pi_{ij}E ((A_i x + B_i u + C_i w)' P_j (A_i x + B_i u + C_i w) x + \rho_j) \biggr]
\end{align*}
The equation seems to miss a + or - before the discounted continuation value. I added a - because there is a minus sign outside the square bracket but I'm really not sure...
->
\begin{align*}
-x' P_i x - \rho_i & = \max_u - \biggl[ x'R_i x + u' Q_i u + 2 u' W_i x \cr &
-\beta \sum_j \Pi_{ij}E ((A_i x + B_i u + C_i w)' P_j (A_i x + B_i u + C_i w) x + \rho_j) \biggr]
\end{align*}