A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide

Cardoulis Laure; Cristofol Michel; Morancey Morgan

doi:10.3934/mcrf.2020054

Article Contents

2021, Volume 11, Issue 4: 965-985. Doi: 10.3934/mcrf.2020054 open access

This issue Previous Article Strict dissipativity analysis for classes of optimal control problems involving probability density functions Next Article

A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

^* Corresponding author
^* Corresponding author

Received: December 31, 2019

Revised: September 30, 2020

Early access: December 2020

Published: December 2021

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Abstract

In this article we consider the inverse problem of determining the diffusion coefficient of the heat operator in an unbounded guide using a finite number of localized observations. For this problem, we prove a stability estimate in any finite portion of the guide using an adapted Carleman inequality. The measurements are located on the boundary of a larger finite portion of the guide. A special care is required to avoid measurements on the cross-section boundaries which are inside the actual guide. This stability estimate uses a technical positivity assumption. Using arguments from control theory, we manage to remove this assumption for the inverse problem with a given non homogeneous boundary condition.

Keywords:

Coefficient inverse problem,
stability estimate,
heat equation,
unbounded domain,
Carleman estimate,
approximate controllability in regular norms.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35K05.

Citation:

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Figure 1. Representation of $ \Omega_l $ and $ \Omega_L $ in a 3D setting

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Figure 2. Lighted lateral boundary from $ a $

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Figure 3. A neighborhood of the lateral boundary

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