Helped by the context of baby-boom, many nations had to look on the issue of their birth rate. This question from many nations had to limit the right to procreation, to preserve the livable space and avoid the overcrowding.

Since 2015, the journal Population has regularly published thematic chronicles dedicated to the state of knowledge on important global population issues. Previous chronicles have addressed key demographic themes such as the masculinization of births, female genital mutilation, mortality inequalities in low-income countries, and abortion worldwide. This new chronicle is dedicated to demographic aging. While aging is occurring in all countries, the situations vary so much that the challenges are extremely different depending on the country : some countries still have very young populations, such as those in the Global South, unlike the populations of European countries, North America, or Japan, for example. In these developed countries, the process of population aging is already advanced, although the timing and extent of the phenomenon can vary. With continued increases in life expectancy and the arrival of large baby-boom generations at advanced ages, the population is inevitably aging, presenting numerous challenges to society as a whole : individuals, families, institutions, and governments.

We propose to control the population in order to avoid, for example, irregularities in natural resources, overpopulation, in search of an ideal population (so that, for example, pension systems can adapt to the increase in the number of elderly people), etc.

This is a population dynamics model where the control is localized on the non-local term.

The model is following \[\label{} \left\lbrace \begin{array}{ll} \partial_{t}y(x,a,t)+\partial_{a}y(x,a,t)-\triangle y(x,a,t)+\mu(a) y(x,a,t)=0 & \text{ in } \Omega\times(0,A)\times(0,T),\\ \\\partial_{\nu}y(x,a,t)=0 &\text{ in } \partial\Omega\times(0,A)\times(0,T),\\ \\y(x,0,t)=\mathds{1}_{\omega}(x)v(x,t)+\int\limits_{0}^{A}\beta(x,a)y(x,a,t)da &\text{ in } \Omega\times(0,T),\\ \\y(x,a,0)=y_{0}(x,a)&\text{ in } \Omega\times(0,A), \end{array} \right.\] where \(y(x,a,t)\) is a distribution of individuals at location \(x\in\Omega\), of age \(a\in (0,A)\), and at time \(t\in (0,T)\), with \(y_0\) being the initial population. Moreover,

\(\partial_{t}y(x,a,t)+\partial_{a}y(x,a,t)\) represents the transport in age at a constant speed,

\(\mu(a) y(x,a,t)\) the mortality,

\(\triangle y(x,a,t)\) the movement of individuals in \(\Omega,\)

\(\partial_{\nu}y(x,a,t)\) the isolation of the population (no emigration or immigration), and \(y(x,0,t)\) the population renewal equation.

\(\beta(x,a)\) and \(\mu(a)\), representing the fertility and mortality functions respectively, satisfy the following demographic conditions : \[\begin{aligned} (H_1): \left\lbrace\begin{array}{l} \mu\geq 0 \text{ a.e. in } (0,A),\\ \mu\in L^{1}_{loc}(0,A)\quad \displaystyle\int_{0}^{A}\mu(a)da=+\infty, \end{array} \right.\end{aligned}\]

\[\begin{aligned} (H_2):\left\lbrace\begin{array}{l} \beta\in C^{1}(\Omega\times[0,A]) \text{ and }\beta(x,a)\geq 0\text{ }\forall (x,a)\in \Omega\times[0,A],\\ \beta(x,a)=0\quad a\in (0,a_b),\quad 0<a_0\leq a_b<A \end{array} \right.\end{aligned}\] We aim to address the following questions :

When do we have almost certain extinction of the population ?

How can we achieve an ideal population at minimal cost ?

Regarding the null-controllability of the model (1), we construct a null-control feedback by for the following distributed model, \[\left\lbrace \begin{array}{ll} \partial_{t}y(x,a,t)+\partial_{a}y(x,a,t)-\triangle y(x,a,t)+\mu(a) y(x,a,t)=\mathds{1}_{\omega\times[0,a_0]}(x,a)v(x,a,t) &\text{in }\quad \Omega\times(0,A)\times(0,T)\\ \\\partial_{\nu}y(x,a,t)=0 &\text{in }\quad \partial\Omega\times(0,A)\times(0,T)\\ \\y(x,0,t)=\int_{0}^{A}\beta(x,a)y(x,a,t)da &\text{in }\quad \Omega\times(0,T)\\ \\y(x,a,0)=y_{0} &\text{in }\quad \Omega\times(0,A) \end{array} \right.\label{}\] which is already established in by establishing an observability inequality. Then, by studying the behavior of the null-control and the controlled state as \(a_0\) goes to zero, we obtain the null-controllability result of system (1), Theorem 2.1 below.

About the null-controllability of (1), we provide a positive answer at the time \(T>A,\) for every \(y_0\in L^2(\Omega\times(0,A)).\) Indeed,

Consider the assumptions (H1-H2), if \(T>A,\) then for every \(y_0\in L^2(\Omega\times[0,A]),\) there exists \(v_1 \in L^2(\omega\times[0,T])\) such that the solution \(y\) to (1.1) verifies \[\begin{aligned} y(x,a,T)=0\text{ a.e. }x\in\Omega\quad a\in(0,A)\end{aligned}\]

And this is thanks to the strategy used in , with the null-control feedback is \[v_1(x,t)=\left\lbrace \begin{array}{ll} 0\qquad & t-a<0,\\ \\-\int_{t}^A \beta(x,a)\pi(a) \dfrac{e^{t\triangle}y_{0}(x,a-t)}{\pi(a-t)\mathds{1}_{\omega}(x)}da \qquad & t-a>0, \end{array} \right.\] and this controlled state is \[y(x,a,t)=\left\lbrace \begin{array}{ll} \dfrac{\pi(a)}{\pi(a-t)}e^{t\triangle}y_{0}(x,a-t)\qquad & t-a<0,\\ \\0 \qquad & t-a>0. \end{array} \right.\]

To achieve an ideal population at minimal cost, a positive answer is provided thanks to the Turnpike property. A tool for qualitative and quantitative analysis of dynamic systems : on the one hand, it study the tendency of optimal solutions and on the other hand, determine the energy that the system must provide to be on the path of equilibrium.

[] Suppose \(y_0\in L^2(\Omega\times(0,A)),\quad y_d\in L^2(\Omega\times[0,A])\) are fixed and let us the system (1) null-controllable in some time \(T>A\) in the sense of Theorem 2.1. There exists a couple of positive constants \((C,\lambda),\) independent of \(y_0\) and \(y_d\), such that for any large enough \(T > 0\), the unique solution \((y_T,v_T)\) of a dynamic optimal control problem subject to (1) satisfies \[\begin{aligned} \label{eq1.7} \Vert y_T(t)-\bar{y}\Vert_{L^2(\Omega\times[0,A])}+\Vert v_T(t)-\bar{v}\Vert_{L^2(\omega)}\leq C\left(\Vert y_0 -\bar{y}\Vert e^{-\lambda t}+\Vert p_T(T)-\bar{p}\Vert e^{-\lambda (T-t)}\right)\end{aligned}\] for almost every \(t\in [0,T],\) where \((\Bar{y},\Bar{v})\) denotes the unique solution to a steady problem subject to static equation of (1) and \((p_T,\Bar{p})\) are optimal dynamic and steady adjoint state.

The inequality (6) shows us the energy that the dynamic system must provide to be on the turnpike in order to reach the running target \(y_d.\)

The Turnpike property can be used, for example, in ecology to manage animal and plant populations, ensuring a sustainable balance between species.