  In this paper we consider the numerical approximation of infinite horizon
problems via the dynamic programming approach. The value function of the
problem solves a Hamilton-Jacobi-Bellman (HJB) equation that is approximated by
a fully discrete method. It is known that the numerical problem is difficult to
handle by the so called curse of dimensionality. To mitigate this issue we
apply a reduction of the order by means of a new proper orthogonal
decomposition (POD) method based on time derivatives. We carry out the error
analysis of the method using recently proved optimal bounds for the fully
discrete approximations. Moreover, the use of snapshots based on time
derivatives allow us to bound some terms of the error that could not be bounded
in a standard POD approach. Some numerical experiments show the good
performance of the method in practice.
