I am trying to solve the following Bellman Equation:

$V(s) =\max_u \left[a'u - (u-s)'Q(u-s) + V(u)\right]$

In the equation above, $s,u,a\in \mathbb R^n$, $Q\in \mathbb R^{n\times n}$ is positive semidefinite. The equation above is very similar to the solution equation for an infinite-horizon, discrete-time Linear-Quadratic Regulator Problem. It can be restated as

$V(s) =\max_u \left[a'(u+s) - u'Qu + V(s+u)\right]$

However, there are linear terms in $s$ and $x$. My question is whether anyone knows of a reference in which the LQR problem is studied in the case of linear terms in the cost function.