Controllability for a population equation with interior degeneracy

TL;DR

This paper investigates the null controllability of a population model with interior degeneracy, establishing Carleman estimates and control existence in different spatial regions, thus advancing control theory for degenerate PDEs.

Contribution

It provides new Carleman estimates and null control results for a population PDE with interior degeneracy, extending previous work in the field.

Findings

Carleman estimates for the adjoint problem were established.

Null control functions were constructed for different control regions.

The results complete the understanding of controllability for this class of degenerate models.

Abstract

We deal with a degenerate model in divergence form describing the dynamics of a population depending on time, on age and on space. We assume that the degeneracy occurs in the interior of the spatial domain and we focus on null controllability. To this aim, first we prove Carleman estimates for the associated adjoint problem, then, via cut off functions, we prove the existence of a null control function localized in the interior of the space domain. We consider two cases: either the control region contains the degeneracy point $x_0$, or the control region is the union of two intervals each of them lying on one side of $x_0$. This paper complement some previous results, concluding the study of the subject.