Null controllability of a nonlinear age and two-sex population dynamics structured model

TL;DR

This paper investigates the null controllability of a nonlinear age-structured two-sex population model, focusing on extinction scenarios for males and females using observability inequalities, linear control, and fixed point methods.

Contribution

It introduces a novel approach combining observability, linear controllability, and fixed point techniques to establish null controllability for a nonlinear population model.

Findings

Achieved null controllability results for both sexes under certain conditions.

Established a method using observability inequalities and fixed point theorems.

Provided conditions for total population extinction within a specified time frame.

Abstract

This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at birth level. \noindent In this work we consider two cases of null controllability problem. \noindent The first problem is related to the extinction of male and female subpopulation density. \noindent The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if $A$ is the maximal age expectancy, a time interval of duration $A$ after the extinction of males or females, one must get the total extinction of the population. \noindent Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system and after the Kakutani's fixed point theorem.