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：本文研究通过动态规划方法对无限时域问题进行数值逼近。该问题的值函数满足Hamilton-Jacobi-Bellman（HJB）方程，我们采用全离散格式进行近似。众所周知，此类数值问题受困于所谓的"维数灾难"。为缓解此问题，我们提出一种基于时间导数的新型本征正交分解（POD）方法实现降阶处理。利用近期证明的全离散逼近最优界，我们对该方法进行了误差分析。此外，基于时间导数的快照使用使得我们能够界定标准POD方法中无法界定的若干误差项。数值实验表明该方法在实际应用中具有良好的性能表现。