On uniform null controllability of transport-diffusion equations with vanishing viscosity limit

TL;DR

This paper investigates the uniform null controllability of transport-diffusion equations as viscosity vanishes, establishing conditions under which control costs remain bounded or explode exponentially, depending on velocity trajectories.

Contribution

It provides new results on the control cost behavior for transport-diffusion equations with vanishing viscosity, addressing an open problem from 2007.

Findings

Control cost remains bounded for small viscosity and large control time when trajectories enter the control region.

Control cost explodes exponentially as viscosity approaches zero when trajectories do not enter the control region.

Results apply to cases where velocity is the gradient of a scalar field and more general velocity fields.

Abstract

This paper aims to address an interesting open problem, posed in the paper "Singular Optimal Control for a Transport-Diffusion Equation" of Sergio Guerrero and Gilles Lebeau in 2007. The problem involves studying the null controllability cost of a transport-diffusion equation with Neumann conditions, where the diffusivity coefficient is denoted by $\varepsilon>0$ and the velocity by $\mathfrak{B}(x,t)$. Our objective is twofold. First, we investigate the scenario where each velocity trajectory $\mathfrak{B}$ originating from $\overline{\Omega}$ enters the control region in a shorter time at a fixed entry time. By employing Agmon and dissipation inequalities, and Carleman estimate in the case $\mathfrak{B}(x,t)$ is the gradient of a time-dependent scalar field, we establish that the control cost remains bounded for sufficiently small $\varepsilon$ and large control time. Secondly, we explore the case where at least one trajectory fails to enter the control region and remains in $\Omega$. In this scenario, we prove that the control cost explodes exponentially when the diffusivity approaches zero and the control time is sufficiently small for general velocity.