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].

Due to the presence of the transport phenomena, the system \eqref{eq:main} is not null controllable for small time (see for instance [2, Proposition 5.1]). This can also be seen from the following numerical simulation. Let us consider the case $A = 0$ and $B = 1,$ i.e the classical Lotka-McKendrick system. We choose $a_1 = 0,$ $a_2 = .2$ and $a_{\dagger} = 2.$ We choose a initial data $q_0$ for the system \eqref{eq:adj}, localized in a neighbourhood of $a = 1.8,$ and we plot the solution of the adjoint problem \eqref{eq:adj}(See Fig 3). We see that, the solution propages along the characteristic lines. It reaches the observation/control region $(a_1, a_2) = (0,.2)$ in time $t > 1.8.$ Before this time the $L^2$ norm of the solution over the obsevation region is small. Hence the lack of null controllability for small time.

In this video we see the evolution of the adjoint system \eqref{eq:adj}. We notice the same behavious described above.

## 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.