On Uniform Null Controllability of Transport–Diffusion Equations With Vanishing Viscosity Limit

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 $\epsilon >0$ and the velocity by $B(x,t)$. Our objective is twofold. First, we investigate the scenario where each velocity trajectory $B$ originating from $\bar{\Omega }$ 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 $B(x,t)$ is the gradient of a time-dependent scalar field, we establish that the control cost remains bounded for sufficiently small $\epsilon$ 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.