Controllability results for cascade systems of $m$ coupled $N$-dimensional Stokes and Navier-Stokes systems by $N-1$ scalar controls

TL;DR

This paper establishes the null-controllability of cascade systems of coupled Stokes and Navier-Stokes equations using scalar controls, employing Carleman estimates and fixed point techniques.

Contribution

It introduces a novel controllability result for coupled fluid systems with scalar controls acting on only one component, extending previous control theory results.

Findings

Null-controllability of coupled Stokes systems proven

Local null-controllability of Navier-Stokes systems demonstrated

Control achieved with N-1 scalar controls acting on one system

Abstract

In this paper, we deal with the controllability properties of a system of $m$ coupled Stokes systems or $m$ coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a $m\times N$ state (corresponding to the velocities of the fluids) by $N-1$ distributed scalar controls. The proof of the controllability of the coupled Stokes system is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.