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

TL;DR

This paper investigates the null-controllability cost of tangential transport-diffusion equations with vanishing viscosity, revealing bounded costs under certain conditions and exponential blow-up when trajectories do not enter the control region.

Contribution

It provides new results on control cost behavior for transport-diffusion equations, addressing an open problem and analyzing different trajectory scenarios with advanced inequalities.

Findings

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

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

Employs Agmon inequalities and Carleman estimates to establish controllability results.

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. Our objective is twofold. Firstly, we investigate the scenario where each trajectory of the tangential velocity enters the control region in a shorter time at a fixed entry time. By employing Agmon inequalities and Carleman estimates, we establish that the control cost remains bounded for sufficiently small diffusivity and large control time. Secondly, we explore the case where at least one trajectory fails to enter the control region. In this scenario, we prove that the control cost explodes exponentially when the diffusivity approaches zero and the control time is sufficiently small.