The POD Dirichlet Boundary Control of the Navier-Stokes Equations: A Low-dimensional Approach to Optimal Control with High Smoothness

Abstract

The proper orthogonal decomposition(POD) is an approach to capture a reduced order basis functions for a dynamical system. Utilizing the order reduction property of POD basis for minimizing computational cost to unsteady fluid flow control problem, we present a POD-based framework of the unsteady Dirichlet boundary control problem for Navier-Stokes equations. An extra basis function can be therefor constructed and appended into the general POD subspace, which as a key step enables the POD approach to the Dirichlet boundary control and results in the control problem merely in time scale. In the paper the excellent quality and flexibility of the POD approach to Dirichlet boundary flow control are confirmed numerically in several flow matching control examples.