An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

TL;DR

This paper introduces an efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow, combining model reduction via Krylov subspaces with LQR control to stabilize complex fluid dynamics systems.

Contribution

The paper develops a novel projection-based algorithm using extended block Krylov subspaces for model reduction and applies Riccati feedback for stabilization of Navier-Stokes systems.

Findings

The proposed method effectively stabilizes large-scale Navier-Stokes systems.

Numerical results show improved performance over existing methods.

The approach reduces computational complexity in feedback stabilization.

Abstract

Navier-Stokes equations are well known in modelling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performances of our proposed method compared to other known methods.