Work Packages Models involving memory terms and hybrid PDE+ODE systems

Controllability of a Class of Infinite Dimensional Systems with Age Structure

Author: - 07 June 2019

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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) $\displaystyle \int_0^{a_\dagger} \mu(a) \ {\rm d} a = \infty.$
Before we state our result, let us introduce the notion of null controllability of the pair $(A,B).$
We say that a pair $(A,B)$ is null-controllable in time $\tau,$ if for every $z_{0} \in X$ there exists a control $u \in L^{2}(0,\tau,U)$ such that, the solution of the system \begin{equation*} \dot z(t) = A z(t) + Bu(t) \quad t \in [0,\tau], \qquad z(0) = z_{0}, \end{equation*} satisfies $z(\tau) = 0.$
We prove the following result
Assume that $\beta$ and $\mu$ satisfy the conditions (H1)-(H3) above. Moreover, suppose that the fertility rate $\beta$ is such that \begin{equation} \label{eq:beta} \beta(a)= 0 \mbox{ for all } a \in (0,a_{b}), \end{equation} for some $a_b\in (0,a_\dagger)$ and that $a_1 < a_b$. Let us assume that the pair $(A,B)$ is null controllable in arbitrary time. Then for every $\tau > a_{1} + a_{\dagger}-a_{2}$ and for every $p_{0} \in \chi$ there exists a control $v \in L^{2}(0,\tau;\mathcal{U})$ such that the solution $p$ of \eqref{eq:main} satisfies \begin{equation} p(\tau,a) = 0 \mbox{ for all } a \in (0,a_\dagger). \end{equation}
It is well known that null controllability is equivalent to final state observability of the adjoint system (see for instance [4, Section 11.2]). The adjoint of the above system reads as \begin{equation} \label{eq:adj} \begin{cases} \displaystyle \frac{\partial q}{\partial t} - \frac{\partial p}{\partial a} - A^* p + \mu(a) p - \beta(a) q(t,0) = 0, & t \geqslant 0, a \in (0,a_\dagger), \\ \displaystyle q(t,a_\dagger) = 0, & t \geqslant 0, \\ \displaystyle q(0,a) = q_{0}, \end{cases} \end{equation} where $A^*$ is the adjoint of $A.$ In order to prove Theorem 1, it is enough to prove the following
Let us assume the hypothesis of Theorem 1. Then for every $\tau > a_1 + a_\dagger - a_2$ there exists $k_\tau > 0$ such that the solution $q$ of \eqref{eq:adj} satishfies \begin{equation} \label{eq:est-adj} \int_0^{a_\dagger} \|q(\tau, a)\|^2_X \ da \leqslant k_\tau^2 \int_0^\tau \int_{a_1}^{a_2} \left\|B^*q(t,a) \right\|^2_U \ da dt, \qquad \qquad (q_0 \in \mathcal{X}). \end{equation}
Let us give an idea of the proof. We combine characteristics method with final state observability of the pair $(A^*, B^*).$ For siplicity in the presentation, let us assume that $\mu = 0,$ $a_1 = 0,$ $a_2 < a_b$ and $\tau = 2a_\dagger.$ The detail proof of Theorem 1, in a slightly more general case, can be found in [3]. Integrating along the characteristic lines, it is not very difficult to see that \begin{equation*} q(t,a) = \int_{t + a - a_\dagger}^{t} e^{(t-s)A^*} \beta(a+ \tau - s) q(s, 0) \ ds, \qquad t > a_\dagger. \end{equation*} Therefore, \begin{equation} \label{est0} \int_0^{a_\dagger} \|q(\tau, a)\|^2_X \ da \leqslant C \int_{a_\dagger}^\tau \left\|q(t,0) \right\|^2_X \ dt. \end{equation} Thus to prove \eqref{eq:est-adj}, we need to estimate the right hand side of the above estimate. We now make use of the condition \eqref{eq:beta}. Note that, due to this condition, $q$ satisfies \begin{equation} \label{eq:adj-2} \displaystyle \frac{\partial q}{\partial t} - \frac{\partial q}{\partial a} - A^* p + \mu(a) p = 0, \quad t \geqslant 0, a \in (0,a_2). \end{equation} For a.e. $t \in (a_\dagger, \tau),$ we define $w(s) = q(s, t- s),$ $s \in (t - a_2, t).$ Then $w$ solves \begin{equation*} \displaystyle \frac{\partial w}{\partial s} - A^* w = 0, \quad s \in (t - a_2, t). \end{equation*} Since the pair $(A, B)$ is null controllable in any time, or equivalently the pair $(A^*, B^*)$ is final-state observable in any time, there exists a constant $C > 0,$ depending only on $a_2, A$ and $B$ such that \begin{equation*} \left\| w(t) \right\|_{X}^2 \leqslant C \int_{t-a_2}^t \left\| B^* w(s) \right\|^2_U \ ds. \end{equation*} Coming back to $q$ and integrating over $[a_\dagger, \tau]$ with respect to $t,$ we have that \begin{equation} \int_{a_\dagger}^\tau \|q(t,0)\|^2_X \ dt \leqslant \int_{0}^\tau \int_{0}^{a_2} \left\|B^* q(t,a)\right\|^2 \ da dt. \end{equation} Combining the above estimate together with \eqref{est0}, we get \eqref{eq:est-adj}. In the two figures below, we ilustrate the estimate of $q(t,0)$ and the final-state observability of $q$ in the general case.

trace
Fig.1 - An ilustration of the estimate of $q(t,0).$ We apply final state observability of $(A^*, B^*)$ along the characteristics. Since we want to estimate $q(t,0),$ we consider the trajectory $\gamma(s) = (t-s,s),$ $s \leqslant t \leqslant \tau$ (or equivalently the backward characteristics starting from $(t,0).$)
final
Fig.2 - An illustration of the final-state observability at time $\tau > a_1 + a_\dagger - a_2$: For $ a \in (0,a_2 - \varepsilon)$ (blue region) the backward characteristics starting from $t = \tau$ enters the observation domain. For $a \in (a_2 - \varepsilon, a_\dagger)$, the backward characteristics (green region) hits the line $a = a_\dagger$, gets renewed by the renewal condition $\beta(a)q(t,0)$ and then enters the observation domain (purple region).

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.

f1
f2
Fig.3 : Solution of the adjoing system \eqref{eq:adj}, localized at $t = 0$ near $a = 1.8,$ that propagates with the velocity $1$ to the left. As soon as it reaches $a =0,$ it gets renewed by the renewal condition and propagates to the left again.

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.