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考虑无限时域问题的数值近似。该问题的值函数定义了Hamilton-Jacobi-Bellman (HJB)方程，并通过全离散方法进行近似。众所周知，数值问题因所谓的维数灾难而难以处理。为缓解此问题，我们采用一种基于时间导数的新的本征正交分解（POD）方法进行降阶。利用近期证明的全离散近似最优界，我们对方法进行了误差分析。此外，基于时间导数的快照集的使用使我们能够界定额外误差项，这些项在标准POD方法中无法被界定。数值实验展示了该方法在实际应用中的良好性能。