Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations

TL;DR

This paper develops a finite-dimensional feedback control method to stabilize a non-stationary solution of the 3D Navier-Stokes equations, using observability inequalities and control theory techniques.

Contribution

It introduces a novel control approach that stabilizes the full Navier-Stokes system around a non-stationary solution in three dimensions.

Findings

Constructed a finite-dimensional feedback control for stabilization.

Proved local stabilization of the nonlinear Navier-Stokes system.

Utilized observability inequalities and regularization properties.

Abstract

We consider the Navier-Stokes system in a bounded domain with a smooth boundary. Given a sufficiently regular time-dependent global solution, we construct a finite-dimensional feedback control that is supported by a given open set and stabilizes the linearized equation. The proof of this fact is based on a truncated observability inequality, the regularizing property for the linearized equation, and some standard techniques of the optimal control theory. We then show that the control constructed for the linear problem stabilizes locally also the full Navier-Stokes system.