Approximation of Optimal Control Problems for the Navier-Stokes equation via multilinear HJB-POD

TL;DR

This paper introduces a novel method for approximating optimal control problems for the Navier-Stokes equations using multilinear HJB-POD, effectively addressing high-dimensional challenges with dynamic programming and numerical examples.

Contribution

It presents a new approach combining multilinear approximation with dynamic programming for high-dimensional Navier-Stokes control problems, diverging from traditional Pontryagin-based methods.

Findings

Effective mitigation of the curse of dimensionality.

Successful numerical examples on classical control problems.

Demonstration of the method's applicability to complex fluid dynamics control issues.

Abstract

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.