Feedback boundary stabilization to trajectories for 3D Navier-Stokes equations

TL;DR

This paper develops a finite-dimensional boundary feedback controller that stabilizes the 3D Navier-Stokes equations around a nonstationary trajectory, supporting the boundary control in a specified region.

Contribution

It introduces a novel boundary feedback control method for stabilizing 3D Navier-Stokes trajectories, extending linear stabilization techniques to nonlinear systems.

Findings

Controller stabilizes the linear Oseen-Stokes system around the trajectory.

Controller also stabilizes the nonlinear Navier-Stokes system locally.

Supports boundary control in a specified boundary subset.

Abstract

Given a nonstationary trajectory of the Navier-Stokes system, a finite-dimensional feedback boundary controller stabilizing locally the system to the given trajectory is derived. Moreover the controller is supported in a given open subset of the boundary of the domain containing the fluid. In a first step a controller is derived that stabilizes the linear Oseen-Stokes system "around the given trajectory" to zero; for that a corollary of a suitable truncated boundary observability inequality, the regularizing property for the system, and some standard techniques of the optimal control theory are used. Then it is shown that the same controller also stabilizes, locally, the Navier-Stokes system to the given trajectory.