Insensitizing controls for the Navier-Stokes equations

TL;DR

This paper establishes the existence of insensitizing controls for the Navier-Stokes equations, ensuring the $L^2$-norm of solutions in a subset remains unaffected by data variations, using Carleman inequalities and controllability techniques.

Contribution

It introduces a novel approach to insensitizing controls for Navier-Stokes equations via Carleman inequalities and cascade system controllability, extending control theory in fluid dynamics.

Findings

Existence of insensitizing controls for Navier-Stokes equations.

Global Carleman inequality for linearized Navier-Stokes system.

Null controllability results for cascade systems.

Abstract

In this paper, we deal with the existence of insensitizing controls for the Navier-Stokes equations in a bounded domain with Dirichlet boundary conditions. We prove that there exist controls insensitizing the $L^2$ -norm of the observation of the solution in an open subset $\mathcal{O}$ of the domain, under suitable assumptions on the data. This problem is equivalent to an exact controllability result for a cascade system. First we prove a global Carleman inequality for the linearized Navier-Stokes system with right-hand side, which leads to the null controllability at any time $T>0$. Then, we deduce a local null controllability result for the cascade system.