This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is:
$k_{t+1}=\beta\alpha k_t^\alpha$,
where $\alpha,\beta \in R(0,1)$
now I want to find out if this expression converges to a steady state, defined as
$\lim_{t\to\infty} k_{t+1}=k_t=k^*.$
I thought about trying to tackle it as an eigenvalue problem, of the form
$\left[ \begin{matrix} \log k(t+1)\\ \log k(t+2) \end{matrix} \right] = \left[ \begin{matrix} 0 &1\\ 0 &\alpha \end{matrix}\right] \left[ \begin{matrix} \log k(t)\\ \log k(t+1)\end{matrix} \right] + \left[ \begin{matrix} 0\\ \log\beta + \log\alpha \end{matrix} \right]$
or
$U_{t+1} = AU_t +B$
My question is this: will the system converge as long as the eigenvalues of A are inside the unit circle? I'm concerned about how the contribution of B behaves over time. Am I on the right track with this thing at all?