Infinite dimensional dynamical systems coupling age structuring with diffusion appear naturally in population dynamics, medicine or epidemiology. A by now classical example is the Lotka-Mckendrick system with spatial diffusion. The aim of this blog is to study contaollability properties of such age structured models in an unified manner.

Let $A : \mathcal{D}(A) \to X$ be the generator of a $C^0$ semigroup $\mathbb{S}$ on the Hilbert space $X$ and let $U$ be another Hilbert space. Both $X$ and $U$ will be identified with their duals. Let $B$ be a (possibly unbounded) linear operator from $U$ to $X$, which is supposed to be an admissible control operator for $\mathbb{S}$. In the examples we have in mind, the above spaces and operators describe the dynamics of a system without age structure. In particular, $X$ is the state space and $U$ is the control space. The corresponding age structured system is obtained by first extending these spaces to \begin{equation} \label{input-space} \mathcal{X} = L^{2}(0,a_{\dagger}; X), \quad \mathcal{U} = L^{2}(0,a_\dagger;U). \end{equation} where $a_\dagger>0$ denotes the maximal age individuals can attain. Let $p(t) \in \chi$ be the distribution density of the individuals with respect to age $a\geqslant 0$ and at some time $t \geqslant 0.$ Then the abstract version of the Lotka-McKendrick system to be considered in this paper writes: \begin{equation} \label{eq:main} \begin{cases} \displaystyle \frac{\partial p}{\partial t} + \frac{\partial p}{\partial a} - A p + \mu(a) p = \mathcal{\chi}_{(a_{1},a_{2})} B u, & t \geqslant 0, a \in (0,a_\dagger), \\ \displaystyle p(t,0) = \displaystyle \int_{0}^{a_\dagger} \beta(s) p(t,s) \; {\rm d}s, & t \geqslant 0, \\ \displaystyle p(0,a) = p_{0}, \end{cases} \end{equation} where $\mathcal{\chi}$ is the characteristic function of the interval $(a_{1}, a_{2})$ with $0 \leqslant a_{1} < a_{2} \leqslant a_\dagger$ and $p_{0}$ is the initial population density. In the above system, the positive function $\mu:[0,a_\dagger] \to \mathbb{R}_{+}$ denotes the natural mortality rate of individuals of age $a.$ We denote by $\beta: [0,a_\dagger] \to \mathbb{R}_{+}$ the positive function describing the fertility rate at age $a.$ We assume that the fertility rate $\beta$ a nd the mortality rate $\mu$ satisfy the conditions

- (H1) $\beta \in L^\infty[0, a_\dagger], \; \beta \geqslant 0$ for almost every $a \in [0,a_\dagger].$
- (H2) $\mu \in L^1_{loc}[0, a_\dagger], \; \mu \geqslant 0$ for almost every $a \in [0,a_\dagger].$
- (H3) $\displaystyle \int_0^{a_\dagger} \mu(a) \ {\rm d} a = \infty.$

## Examples

We consider two examples : 1) The classical Lotka-McKendrick system and 2) Lotka-Mckendrick system with diffusion.

1) * The classical Lotka-McKendrick system *: Let us choose $X = \mathbb{R},$ $A=0$ and $B=1.$ Then the system \eqref{eq:main} reduces
to classical Lotka-McKendrick system. By Theorem 1, this system is null controllable in time $\tau > a_1 + a_\dagger - a_2.$ A
similar result was obtained in [1].

2) * The Lotka-McKendrick system with spatial diffusion * : Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$ and
$\omega \subseteq \Omega.$ Let us consider
\begin{equation*}
X = L^2(\Omega), \quad A = \Delta , \quad \mathcal{D}(A) =
\left\{ f \in H^2(\Omega) \mid \displaystyle \frac{\partial f}{\partial n} = 0\right\}, \quad B = \chi_{\omega}.
\end{equation*}
This corresponds to the Lotka-McKendrick model with spatial diffusion ([2]). It is known that, the pair $(A, B)$ or equivalently
the heat equation with localized interior control, is null controllable in any time. Thus we can apply Theorem 1 to conclude that
the system is null controllable in time $\tau > a_1 + a_\dagger - a_2.$

Several other applications can be found in [3].

## Numerical Simulations

We now present some numerical siimulations of the controlled trajectory. We shall consider the case $A = 0$ and $B = 1,$ i.e the classical Lotka-McKendrick system. We also consider the case $a_1 = 0$ and $a_2 = a_{\dagger} =4,$ i.e., the control acts everywhere with respect to age variable. We present numerical simulations in the following two scenarios : 1) null controllabilty, 2) controllability to a steady state.

#### Null Controllability

By Theorem 1 we know that system (1) is null controllable in any time. In the following video, we see the evolution of unctrolled trajectory and controlled trajectories to zero with controllability time $t=2$ and $t =5.$We now plot the control functions. In the following two videoes, we see the evolution of the control functions at controllability time $t = 2$ and $t =5$ respectively.

#### Controllability to a steady state

If we choose $\beta$ and $\mu$ such that $R = \int_0^{a_\dagger} \beta(a) e^{-\int_0^r \mu(r) \ dr } da = 1,$ then $p_s(a) = \alpha e^{-\int_0^r \mu(r) \ dr }$ with $\alpha \in (0,\infty)$ is a steady state to the system (1) with zero steady control. Thus we can control to these trajectories also. In the following example, we have chosen $a_{\dagger} = 4,$ $a_b = 1,$ $\alpha = 1.25$ and \begin{equation*} \beta(a) = \begin{cases} 0 & \mbox{ if } a \in [0,1), \\ \frac{1}{3}e^{.05 a^2} & \mbox{ if } a \in [1,4], \end{cases} \qquad \mu(a) = .1 a. \end{equation*} In this case we can verify that $R=1$ and $p_s(a) = 1.25 e^{-.05 a^2}.$ As before we take control everywhere, thus the system is controllable to the trajectory $p_s(a)$ in any time. In the following video, we see the evolution of unctrolled trajectory and controlled trajectories to the steady state $p_s(a)$ with controllability time $t=2$ and $t =5.$

As before, in the following two videos we see the evolution of the control functions at time $t =2$ and $t=5$ respectively.

## References

[1] D. Maity, * On the Null Controllability of the Lotka-Mckendrick System,* Submitted.

[2] D. Maity, M. Tucsnak and E. Zuazua, * Controllability and positivity constraints in population dynamics with age
structuring and diffusion *, Journal de Mathématiques Pures et Appliqués. In press, 10.1016/j.matpur.2018.12.006.

[3] D. Maity, M. Tucsnak and E. Zuazua, * Controllability of a Class of Infinite Dimensional
Systems with Age Structure*. Submitted.

[4] M. Tucsnak and G. Weiss, * Observation and control for operator semigroups, *
Birkhäuser Advanced Texts: Basler Lehrbuücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009.