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.

翻译：本文通过动态规划方法研究无限时域问题的数值逼近。该问题的值函数满足Hamilton-Jacobi-Bellman (HJB)方程，采用全离散方法进行逼近。众所周知，此类数值问题因所谓的维数灾难而难以处理。为缓解这一问题，我们提出一种基于时间导数的全新本征正交分解（POD）方法实现降阶。利用近期证明的全离散逼近最优界，我们对所提方法进行误差分析。此外，基于时间导数的快照选取使得标准POD方法中无法界定的某些误差项得以控制。数值实验表明该方法在实际应用中具有良好的性能。