In the Dynamic Programming approach to optimal control problems a crucial role is played by the value function that is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well known that this approach suffers of the "curse of dimensionality" and this limitation has reduced its practical in real world applications. Here we analyze a dynamic programming algorithm based on a tree structure. The tree is built by the time discrete dynamics avoiding in this way the use of a fixed space grid which is the bottleneck for high-dimensional problems, this also drops the projection on the grid in the approximation of the value function. We present some error estimates for a first order approximation based on the tree-structure algorithm. Moreover, we analyze a pruning technique for the tree to reduce the complexity and minimize the computational effort. Finally, we present some numerical tests.

翻译：在动态规划方法中,优化控制问题的最佳控制方法具有关键作用,其价值功能被描述为汉密尔顿-贾科比-贝尔曼(HJB)等式的独特粘度解决方案。众所周知,这一方法受到“维度诅咒”的影响,而这一局限性减少了实际应用的实用性。在这里,我们分析了基于树结构的动态编程算法。树是由时间离散动态构建的,从而避免了使用固定空间网格,而固定空间网格是高维度问题的瓶颈,这也降低了在数值函数近似时对网格的预测。我们为基于树结构算法的第一顺序近似提供了一些误差估计。此外,我们分析了树的裁剪技术,以减少复杂性和尽量减少计算努力。最后,我们提出了一些数字测试。