Time optimal internal controls for the Lotka-McKendrick equation with spatial diffusion

Nicolas Hegoburu

doi:10.3934/mcrf.2019047

Article Contents

2019, Volume 9, Issue 4: 697-718. Doi: 10.3934/mcrf.2019047

This issue Previous Article Backward uniqueness results for some parabolic equations in an infinite rod Next Article On the null controllability of the Lotka-Mckendrick system

Time optimal internal controls for the Lotka-McKendrick equation with spatial diffusion

Nicolas Hegoburu

Institut de Mathématiques de Bordeaux, Université de Bordeaux/Bordeaux INP/CNRS, 351 Cours de la Libération, 33 405 Talence, France

Received: October 2018

Revised: February 2019

Published: November 2019

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Abstract

This work is devoted to establish a bang-bang principle of time optimal controls for a controlled age-structured population evolving in a bounded domain of $ \mathbb{R}^n $. Here, the bang-bang principle is deduced by an $ L^\infty $ null-controllability result for the Lotka-McKendrick equation with spatial diffusion. This $ L^\infty $ null-controllability result is obtained by combining a methodology employed by Hegoburu and Tucsnak - originally devoted to study the null-controllability of the Lotka-McKendrick equation with spatial diffusion in the more classical $ L^2 $ setting - with a strategy developed by Wang, originally intended to study the time optimal internal controls for the heat equation.

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Mathematics Subject Classification: 93B03, 93B05, 92D25, 34K35, 35Q93.

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Figure 1. The spectrum of the free diffusion operator $ A_0 $ (green crosses) and of $ -\Delta $ (red circles)

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