Null Controllability for a Degenerate Structured Population Model

TL;DR

This paper proves null controllability for a complex population model with degenerate diffusion, using innovative methods that avoid traditional Carleman estimates, and provides estimates on control time depending on control support.

Contribution

It introduces a novel approach combining observability estimates and characteristic methods to establish null controllability without relying on Carleman estimates.

Findings

Null controllability is achieved for the degenerate population model.

Control time estimates depend on the support of the control.

The method avoids the use of explicit Carleman estimates.

Abstract

In this paper, we consider the infinite dimensional linear control system describing population models structured by age, size, and spatial position. The diffusion coefficient is degenerate at a point of the domain or both extreme points. Moreover, the control is localized in the space variable as well as with respect to the age and size. For each control support, we give an estimate of the time needed to control the system to zero. We establish the null controllability of the model by using a technique that avoids the explicit use of parabolic Carleman estimates. Indeed, our argument relies on a method that combines final-state observability estimates with the use of the characteristic method.