Birth control and turnpike property of Lotka-McKendrick models with diffusion

TL;DR

This paper investigates the turnpike property in age-structured population models with birth control, establishing null controllability and demonstrating the exponential stability of the system through analytical and numerical methods.

Contribution

It introduces a novel approach to control age-structured populations with birth control, proving null controllability and the turnpike property using advanced control theory techniques.

Findings

Null controllability of the population system is established.

The control operator is shown to be admissible in the state space.

Numerical examples confirm the theoretical results.

Abstract

In this paper, we study the turnpike property in age structured population dynamics with birth control. These models describe the temporal evolution of one or more populations, incorporating age dependence and spatial structure. To this end, we first establish the null controllability of the system: we prove that for any T > A and any initial datum in L2(Omega x (0,A)), the population can be driven to zero using control functions that are spatially localized in time but act only at age a = 0. We then show that although this control is initially applied only at birth, it can be reformulated as a distributed control, and we demonstrate that the resulting control operator is admissible in the state space. Thus, to prove the turnpike property we combine our null controllability results with Phillips' theorem on exponential stability to design a suitable dichotomy transformation based on solutions of the algebraic Riccati and Lyapunov equations. Finally, we present numerical examples that substantiate our analytical findings.