论文标题
通过多连线HJB-POD的Navier-Stokes方程的最佳控制问题近似
Approximation of Optimal Control Problems for the Navier-Stokes equation via multilinear HJB-POD
论文作者
论文摘要
我们考虑通过动态编程方法来考虑Navier-Stokes方程的一些最佳控制问题。这些控制问题在许多工业应用中都出现，从数值的角度来看，这是非常具有挑战性的，因为动力学的半差异对应于非常高维的普通微分方程的进化系统。典型的方法是基于pontryagin的最大原理，并导致两个点边界值问题。在这里，我们基于贝尔曼的价值函数和解决方案提出了一种不同的方法，这是高维度的一个具有挑战性的问题。我们通过在树结构上的动态编程方案以及动态编程方案的最新多线性近似来减轻维度的诅咒。我们讨论了与这种新方法的实施相关的几个方面，并提出了一些数值示例，以说明文献中研究的经典控制问题的结果。
We consider the approximation of some optimal control problems for the Navier-Stokes equation via a Dynamic Programming approach. These control problems arise in many industrial applications and are very challenging from the numerical point of view since the semi-discretization of the dynamics corresponds to an evolutive system of ordinary differential equations in very high dimension. The typical approach is based on the Pontryagin maximum principle and leads to a two point boundary value problem. Here we present a different approach based on the value function and the solution of a Bellman, a challenging problem in high dimension. We mitigate the curse of dimensionality via a recent multilinear approximation of the dynamics coupled with a dynamic programming scheme on a tree structure. We discuss several aspects related to the implementation of this new approach and we present some numerical examples to illustrate the results on classical control problems studied in the literature.
