Controllability of a simplified fluid-structure interaction system

TL;DR

This paper investigates the controllability of a simplified fluid-structure interaction system, demonstrating null-controllability for a coupled Stokes and heat equation model using Carleman estimates, as a step towards more complex models.

Contribution

It introduces a linearized, simplified model coupling Stokes and heat equations and proves its null-controllability, advancing understanding of control in fluid-structure systems.

Findings

The coupled system is null-controllable.

Carleman estimates are effective for this control problem.

The work paves the way for handling more complex structural models.

Abstract

We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.