Controllability of 3D incompressible Euler equations by a finite-dimensional external force

TL;DR

This paper investigates the controllability of the 3D incompressible Euler equations, demonstrating exact controllability in projections and approximate controllability, but not exact controllability with finite-dimensional external forces.

Contribution

It establishes the limits and capabilities of controlling the 3D Euler system using finite-dimensional external forces, including exact and approximate controllability results.

Findings

Velocity and pressure are exactly controllable in projections.

Velocity is approximately controllable.

The system is not exactly controllable with finite-dimensional external forces.

Abstract

In this paper, we study the control system associated with the incompressible 3D Euler system. We show that the velocity field and pressure of the fluid are exactly controllable in projections by the same finite-dimensional control. Moreover, the velocity is approximately controllable. We also prove that 3D Euler system is not exactly controllable by a finite-dimensional external force.