Study of Partial Differential Equation Derived from Population Dynamics with Age Structuring and Diffusion

Abstract

The current research studies the null controllability of an infinite dimensional linear system describing the classical Lotka-McKendrick system of the population dynamics with age structure and spatial diffusion. The active period of ages which we need to control can be small and do not need to apply control for ages a neighbourhood of zero, precisely the active in our case for ages a ∈ (a1, a2) where a1 ∈ [0, aM) and a2 ∈ (a1, aM], before the individuals start to reproduce. We have proven that the population can be driven into zero in a uniform time, excluding some interval of low ages. We proved that by rewriting the system as evolution system on Hilbert space by using semigroup theory to apply a Hilbert Uniqueness Method which is based on Riesz Representation Theorem to find the associate adjoint semigroup to take advantage of the fact that null cotrollability of a linear system is equivalent to the final-state observability inequality of the adjoint system. The controllability of the Lotka-McKendrick equation was achieved by impulsive control by the Event Triggered Method. Moreover, some numerical illustrations were provided to confirm the applicability of our results.