On the cost of null-control of an artificial advection-diffusion problem

TL;DR

This paper investigates how the cost of controlling an artificial advection-diffusion system behaves as viscosity approaches zero, revealing exponential decay or growth depending on control time.

Contribution

It provides a spectral analysis showing the exponential decay or growth of control cost in relation to viscosity and control time.

Findings

Control cost tends to zero exponentially with vanishing viscosity and large control time.

Control cost tends to infinity exponentially with vanishing viscosity and small control time.

Spectral method effectively analyzes null-controllability in high-dimensional systems.

Abstract

In this paper we study the null-controllability of an artificial advection-diffusion system in dimension $n$. Using a spectral method, we prove that the control cost goes to zero exponentially when the viscosity vanishes and the control time is large enough. On the other hand, we prove that the control cost tends to infinity exponentially when the viscosity vanishes and the control time is small enough.