Small-time global exact controllability to the trajectories for the viscous Boussinesq system

TL;DR

This paper establishes the small-time global exact controllability to trajectories for the viscous Boussinesq system in 2D and 3D domains, using boundary control and advanced mathematical techniques.

Contribution

It introduces a novel approach combining domain extension, approximate controllability, and Carleman estimates to achieve controllability results for the viscous Boussinesq system.

Findings

Proves global approximate controllability of the inviscid Boussinesq system.

Establishes local controllability via linearization and Carleman estimates.

Demonstrates controllability with boundary control on small parts of the domain.

Abstract

In this paper, we deal with the global exact controllability to the trajectories of the Boussinesq system. We consider 2D and 3D smooth bounded domains. The velocity field of the fluid must satisfy a Navier slip-with-friction boundary condition and a Robin boundary condition is imposed to the temperature. We assume that one can act on the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following the strategy of Coron et al [J. Eur. Math. Soc., 22 (2020), pp. 1625-1673], which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates.