HJB-POD feedback control for Navier-Stokes equations

TL;DR

This paper introduces a combined POD and Hamilton-Jacobi approach to approximate optimal control for Navier-Stokes equations, demonstrating its effectiveness through numerical tests in 2D.

Contribution

It presents a novel integration of POD model reduction with HJB equations for Navier-Stokes control problems, addressing practical dimension limitations.

Findings

Effective control approximation demonstrated in 2D Navier-Stokes simulations

Convergence of approximation schemes confirmed for any dimension

Numerical results show the method's practical viability

Abstract

In this report we present the approximation of an infinite horizon optimal control problem for the evolutive Navier-Stokes system. The method is based on a model reduction technique, using a POD approximation, coupled with a Hamilton-Jacobi equation which characterizes the value function of the corresponding control problem for the reduced system. Although the approximation schemes available for the HJB are shown to be convergent for any dimension, in practice we need to restrict the dimension to rather small numbers and this limitation affects the accuracy of the POD approximation. We will present numerical tests for the control of the time-dependent Navier-Stokes system in two-dimensional spatial domains to illustrate our approach and to show the effectiveness of the method.