Optimal control problems driven by evolutionary partial differential equations arise in many industrial applications and their numerical solution is known to be a challenging problem. One approach to obtain an optimal feedback control is via the Dynamic Programming principle. Nevertheless, despite many theoretical results, this method has been applied only to very special cases since it suffers from the curse of dimensionality. Our goalis to mitigate this crucial obstruction developing a new version of dynamic programming algorithms based on a tree structure and exploiting the compact representation of the dynamical systems based on tensors notations via a model reduction approach. Here, we want to show how this algorithm can be constructed for general nonlinear control problems and to illustrate its performances on a number of challenging numerical tests. Our numerical results indicate a large decrease in memory requirements, as well as computational time, for the proposed problems. Moreover, we prove the convergence of the algorithm and give some hints on its implementation

翻译：由演化偏微分方程驱动的最优控制问题出现在许多工业应用中，其数值求解被认为是一个具有挑战性的问题。通过动态规划原理可以获得最优反馈控制。然而，尽管有众多理论成果，该方法因受制于维数灾难而仅被应用于非常特殊的情形。我们的目标是缓解这一关键障碍，开发一种基于树结构并利用张量符号表示动态系统的紧凑表示（通过模型降阶方法）的新型动态规划算法。本文旨在展示该算法如何适用于一般非线性控制问题，并通过一系列具有挑战性的数值试验说明其性能。数值结果表明，对于所提出的问题，内存需求和计算时间均大幅减少。此外，我们证明了该算法的收敛性，并给出了其实现的一些要点。