Finite-Dimensional MOR-Based RHC for Steering 2D Navier-Stokes Equations to Desired Trajectories

TL;DR

This paper develops a finite-dimensional model predictive control method for stabilizing 2D Navier-Stokes flows to desired trajectories, combining theoretical analysis with reduced-order modeling for computational efficiency.

Contribution

It introduces a novel RHC scheme with finite-dimensional control and a POD-based model reduction, demonstrating stability and efficiency for complex flow configurations.

Findings

Proven local exponential stabilization of 2D Navier-Stokes equations.

Reduced-order RHC maintains stabilization with lower computational cost.

Numerical experiments confirm theoretical results and practical effectiveness.

Abstract

This paper investigates the local exponential stabilization of the two-dimensional Navier--Stokes equations to a given reference trajectory by means of receding horizon control (RHC). The control is realized as a linear combination of finitely many actuators, represented by indicator functions supported on subsets of a prescribed control subdomain. We establish local exponential stabilizability and suboptimality for the resulting RHC scheme. Numerical experiments for two flow configurations of increasing complexity illustrate the theoretical findings and assess the practical performance of the method. In addition, we propose a model-order-reduced RHC approach based on proper orthogonal decomposition, which significantly reduces the computational cost while maintaining favorable closed-loop stabilization performance in the numerical experiments.