Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

Jon Asier Bárcena-Petisco

doi:10.1051/cocv/2021103

Abstract

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time T sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time T > 0 the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.

Highlights

The transport-diffusion equation with vanishing diffusivity is used to describe the dynamics of some physical and biological phenomena. This is the case of fluid dynamics, as it is explained in Chapter 3 of [1] and the reference therein
In this paper we study the cost of null controllability for transport-diffusion equations with Neumann, Robin and mixed boundary conditions
We show some examples of Neumann and mixed boundary conditions in which for any time T > 0 the cost explodes exponentially as the diffusion coefficient ε vanishes

Summary

Introduction

The transport-diffusion equation with vanishing diffusivity is used to describe the dynamics of some physical and biological phenomena. In this paper we study the cost of null controllability for transport-diffusion equations with Neumann, Robin and mixed boundary conditions. This paper follows a well-established research line which inquires about the cost of the null controllability of systems with a small diffusion and a transport term The first of such control problems was the heat equation in dimension one with Dirichlet boundary conditions in [14]. The current study of (1.1) is a contribution to the literature as it seeks to understand the evolution of the cost of the null controllability of the transport-diffusion equation with a large variety of boundary conditions when the diffusivity vanishes (and, in particular, the case aε = 0 and Γ = ∅ was an open problem proposed in Remark 3 of [24])

Quantification of the main results

The observability problem and the symmetrized system

The regime in which the cost of the control decays exponentially

Open problems