Null Controllability of a Nonlinear Age, Space and Two-sex Structured Population Dynamics Model

Spain 05.08.2022

Null Controllability of a Nonlinear Age, Space and Two-sex Structured Population Dynamics Model

Author: Yacouba Simpore

Motivation and description of two sex structured population dynamics model

Motivation

Malaria is a disease caused by parasites of the genus Plasmodium. According to the WHO, this disease causes approximately one million victims per year worldwide.

Figure 1. Anophel Gambiae

The parasite is transmitted to humans through the bite of an infected mosquito. These mosquitoes, “vectors” of malaria, all belong to the genus Anopheles.

In the fight against malaria and other diseases transmitted to humans by insects (such as the Anopheles Gambiae), many methods are used around the world (development of vaccines, etc.). In recent years, the mosquito control strategy is increasingly used in many countries such as the United States and West Africa.

[1] We have in west Africa “Target Malaria” project underway and which aims to drive the density of wild female mosquitoes to zero in long time horizon. The strategy involves three stages:
• The first phase: Creation of sterile male mosquitoes. The researchers generated male Anopheles gambiae mosquitoes, which can reproduce with wild females, but which will not produce any offspring. This first release represents a test phase which will make it possible to acquire knowledge and build an operational capacity.
• The second phase: (Creation of self-limiting male mosquitoes) consists of releasing male mosquitoes leading to a distortion of the sex ratio of the targeted mosquito population. The purpose of this phase is to learn more about the development of their technology, using a line of self-limiting male mosquitoes, which will decline in number with each generation.
• The third stage or Gene Drive consists of the production of genetically modified, fertile mosquitoes, which can transmit a genetic modification to their offspring. This will be passed on to the following generations. They are currently investigating several options, the two most promising are :
– A genetically modified strain with fertile males that produce predominantly male offspring, leading to a distortion in the sex ratio of the targeted mosquito population;
– A genetically modified strain with fertile males carrying a gene that will spread through the mosquito population and cause females that inherit the gene from both parents to be sterile.

 
[2] In the coming months more than 2 million genetically modified mosquitoes will be released in Florida. The mosquitoes, created by biotech firm Oxitec, will be non-biting Aedes aegypti males engineered to only produce viable male offspring, per the company. Oxitec says the plan will reduce numbers of the invasive Aedes aegypti, which can carry diseases like Zika, yellow fever and dengue.
• In this blog we give mathematically some ideas on the possibility of controlling of mosquitoes population dynamics. For reasons like as the difference in lifespan between male mosquitoes (14 days) and female (30 days) and the difference in mortality functions, we preferred to work with the two-sex model which seems the best fit.
• In the strategies cited, the control methods used seem to be birth control or the combination of birth control and distributed control, in this first work, we will focus on distributed controls.

 

Description of two sex structured population dynamics model

We denote by $\Xi=\omega\times(a_1,a_2)\times (0,T)\subset Q$ and $\Xi'=\omega'\times(b_1,b_2)\times (0,T)\subset Q$ where $Q=\Omega\times(0,A)\times(0,T).$ We denote also $\Sigma=\partial\Omega\times(0,A)\times(0,T)\hbox{, }Q_T=\Omega\times(0,T)$ and $Q_A=\Omega\times(0,A).$

Let $(m,f)$ solution of the following system :

$\left\lbrace \begin{array}{ll} \dfrac{\partial m}{\partial t}+\dfrac{\partial m}{\partial a}-K_m\Delta m+\mu_m m=\chi_{\Xi}v_m&\text{ in }Q,\\ \dfrac{\partial f}{\partial t}+\dfrac{\partial f}{\partial a}-K_f\Delta f+\mu_f f=\chi_{\Xi'}v_f&\text{ in }Q,\\ m(\sigma,a,t)=f(\sigma,a,t)=0&\text{ on }\Sigma, \\ m(x,a,0)=m_0\quad f(x,a,0)=f_0&\text{ in }Q_A,\\ m(x,0,t)=(1-\gamma)N(x,t)\text{, }f(x,0,t)=\gamma N(x,t)& \text{ in }Q_T,\\ N(x,t)=\int_{0}^{A}\beta(a,M)fda\text{; } M=\int_{0}^{A}\lambda(a)mda & \text{ in }Q_T. \end{array} \right. (1.1)$

where $m_0\in L^2(Q_A)$, $f_0\in L^2(Q_A)$, $v_m\in L^2(Q)$, $v_f\in L^2(Q)$ and $\gamma\in(0,1).$

The functions, $m(x,a,t)$ and $f(x,a,t)$ represent the density of males and females of age $a$ at time $t$ in position $x,$ respectively.
We assume that the fertility functions $\beta$, $\lambda$ and mortality $\mu_m$ and $\mu_f$ satisfy the following demographic properties:

$(H_1): \left\lbrace \begin{array}{ll} (i)\hbox{ }\mu_m\geq 0, \quad \mu_f\geq 0\text{ a.e. in } [0,A], \\\ (ii)\hbox{ }\mu_m\in L_{loc}^{1}\left([0,A)\right),\quad \mu_f\in L_{loc}^{1}\left([0,A)\right), \\ (iii)\hbox{ }\int_{0}^{A}\mu_m(a)da=+\infty,\quad \int_{0}^{A}\mu_f(a)da=+\infty. \end{array} \right.$

The functions
$\Pi_{m}(a)=e^{-\int\limits_{0}^{a}\mu_m(s)ds}\text{ and }\Pi_{f}(a)=e^{-\int\limits_{0}^{a}\mu_f(s)ds}$ denote the probability of survival of male individuals of age $a$ and female individuals of age $a,$ respectively.

$(H_{2}): \left\lbrace \begin{array}{ll} (i)\hbox{ }\beta\in C\left([0,A]\times \mathbb{R}\right),\\ (ii)\hbox{ }\beta(a,p)\geq 0\text{ for all } (a,p)\in[0,A]\times \mathbb{R},\\ (iii)\hbox{ }\beta(a,0)=0\text{ in } (0,A). \end{array} \right.$

 

$(H_{3}): \left\lbrace \begin{array}{ll} \lambda\in C^{1}\left([0,A]\right),\\ \lambda\geq 0\text{ for all } a\in [0,A]. \end{array} \right.$

Moreover, we suppose that:

$(H_{4}):\left\lbrace \begin{array}{ll} (i)\hbox{ }\text{there exists }b\in (0,A)\text{ such that }\beta(a,p)=0, \forall (a,p)\in [0,b)\times\mathbb{R},\\ (ii)\hbox{ }\text{ there exists } L>0 \text{ such that }|\beta(a,p)-\beta(a,q)|\leq L|p-q|\\ \text{ for all } p, q \in\mathbb{R}\text{, } a\in [0,A],\\ (iii)\hbox{ }\text{there exists }\beta_{0}>0 \text{ such that } 0\leq\beta(a,p)\leq \beta_{0}\text{, } \forall (a,p)\in [0,A]\times\mathbb{R}. \end{array} \right. (1.2)$

 

$(H_5):\left\lbrace \begin{array}{ll} \lambda\mu_m\in L^{1}((0,A)).\\ \end{array} \right.$

2 Null controllability

2.1 Null controllablity

We have the following

Theorem 2.1.

Suppose that the assumptions $(H_1)-(H_2)-(H_3)-(H_4)-(H_5)$ hold. If $(0,b)\cap(a_1,a_2)\cap(b_1,b_2)\neq \emptyset,$ for every time $T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\}$ and for every $(m_0,f_0)\in \left(L^2(Q_A)\right)^2,$ there exists $(v_m,v_f)\in L^2(\Xi)\times L^2(\Xi')$ such the solution $(m,f)$ of the system (1.1) verifies:

$m(x,a,T)=0 \hbox{ a.e. } x\in \Omega, \hbox{ } a\in (0,A), (2.1)$

$f(x,a,T)=0 \hbox{ a.e. } x\in \Omega,\hbox{ } a\in (0,A). (2.2)$

Remark 2.1.

Notice that $\omega\cap \omega'$ can be empty

Figure 2.Illustration of the control support in space

The following figure gives an estimate of the time in the case of an extreme method of controllability

Figure 3.The method being to eliminate all male and female individuals whose age is between $a_1$ and $a_2$ ($t_3-t_1=A-a_2+\epsilon+a_1$).In this case, all males and females whose age is greater than $a_2$ can live to the maximum age $A$ and can have offspring which can also live to the age of $a_1$.

2.2 Null controllability of auxiliary system

Let $p$ be a function in $L^{2}(Q_T),$ we define the auxilliary system given by:

$\left\lbrace \begin{array}{ll} \dfrac{\partial m}{\partial t}+\dfrac{\partial m}{\partial a}-K_m\Delta m+\mu_m m=\chi_{\Xi}v&\text{ in }Q ,\\ \dfrac{\partial f}{\partial t}+\dfrac{\partial f}{\partial a}-K_f\Delta f+\mu_f f=\chi_{\Xi'}u&\text{ in }Q,\\ m(\sigma,a,t)=f(\sigma,a,t)=0&\text{ on }\Sigma, \\ m(x,a,0)=m_0\quad f(x,a,0)=f_0 &\text{ in }Q_A,\\ m(x,0,t)=(1-\gamma)\int_{0}^{A}\beta(a,p)fda,\\ f(x,0,t)=\gamma \int_{0}^{A}\beta(a,p)fda& \text{ in }Q_T.\\ \end{array} \right. (2.3)$

The system (2.3) admits a unique solution $(m,f)\in (L^2((0,A)\times (0,T);H^{1}_{0}(\Omega)))^2$ and the system (2.3) is null controllable for every $T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\}.$ Moreover the null controllability of the system (2.3) is equivalent of the Observability inequality. Let $(n,l)$ be the solution of the following adjoint system to the auxilliary system (2.3)

$\left\lbrace \begin{array}{ll} -\dfrac{\partial n}{\partial t}-\dfrac{\partial n}{\partial a}-K_m\Delta n+\mu_m n=0 &\text{ in }Q ,\\ -\dfrac{\partial l}{\partial t}-\dfrac{\partial l}{\partial a}-K_f\Delta l+\mu_f l=(1-\gamma)\beta(a,p)n(x,0,t)+\gamma\beta(a,p)l(x,0,t)&\text{ in }Q,\\ n(\sigma,a,t)=l(\sigma,a,t)=0&\text{ on }\Sigma, \\ n(x,a,T)=n_T\quad l(x,a,T)=l_T&\text{ in }Q_A,\\ n(x,A,t)=0,\quad l(x,A,t)=0& \text{ in }Q_T.\\ \end{array} \right. (2.4)$

Under the assumptions on the time $T,$ we have the following:

2.3 Observability Inequality

Theorem 2.2

Under the assumptions of Theorem 1, for every $T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\},$ there exists a constant $C_{T}>0$ independent of $p$ such that the solution $(n,l)$ of the system (2.4) verifies:

$\int_{0}^{A}\int_{\Omega}n^2(x,a,0)dxda+\int_{0}^{A}\int_{\Omega}l^2(x,a,0)dxda$

$\leq C_{T}\left(\int_{\Xi}n^2(x,a,t)dxdadt+\int_{\Xi'}l^2(x,a,t)dxdadt\right).$

 

2.4 Representation for the solution of the adjoint system

The idea to establish the observability inequality is the estimation of the non local terms of the adjoint system. For this reason, we first begun to formulating a representation of the solution of cascade adjoint system by caractheristics method and semigroup.

For $(n_T,l_T)\in(L^2(Q_A))^2,$ under the assumptions $(H_1)$ and $(H_2),$ the cascade system (2.4) admits a unique solution $(n,l).$ Moreover, integrating along the characteristics line the solution $(n,l)$ of (2.4) is given by:

$n(t) = \begin{cases} \dfrac{\pi_1(a+T-t)}{\pi_1(a)}e^{(T-t) K_m\Delta} n_T(x,a+T-t) \text{ if } T-t \leq A-a, \\ 0 \text{ if } A-a \lt T-t, \end{cases} (2.5)$

and

$l(t)= \begin{cases} \scriptstyle \frac{\pi_2(a+T-t)}{\pi_2(a)}e^{(T-t) K_f\Delta}l_T(x,a+t-T)\\ \scriptstyle +\int_{t}^{T}\frac{\pi_2(a+s-t)}{\pi_2(a)}e^{(s-t) K_f\Delta}\beta(a+s-t,p(x,s))\left((1-\gamma)n(x,0,s)+\gamma l(x,0,s)\right)ds\text{ in } D_1, \\ \scriptstyle \int_{t}^{t+A-a}\frac{\pi_2(a+s-t)}{\pi_2(a)}e^{(s-t) K_f\Delta}\beta(a+s-t,p(x,s))\left((1-\gamma)n(x,0,s)+\gamma l(x,0,s)\right)ds \text{ in } D_2, \end{cases} (2.6)$

where $\pi_1(a)=e^{-\int_{0}^{a}\mu_m(r)dr}\text {, }\pi_2(a)=e^{-\int_{0}^{a}\mu_f(r)dr}\text{, } e^{tK_m \Delta}$ is the semigroup of $-K_m\Delta$ with the Dirichlet boundary condition and

$D_1=\{(a,t)\in (0,A)\times (0,T) \text{ such that } T-t\leq A-a\},$

$D_2=\{(a,t)\in (0,A)\times (0,T) \text{ such that } T-t > A-a\}.$

Using the fact that $\beta(a,p)=0\hbox{ for all }a\in[0,b).$ We establish the following:

 

2.4.1 Estimations of the non-local terms

Proposition 2.1.
Under the assumptions of Theorem 1, for every $\eta$ satisfying $a_1 \lt \eta \lt T,$ there exists $C>0$ such that the following inequality

$\int_{0}^{T-\eta}\int_{\Omega}n^2(x,0,t)dxdt\leq C\int_{0}^{T}\int_{a_1}^{a_2}\int_{\omega}n^2(x,a,t)dxdadt (2.7)$

holds.

For every $\eta$ verifying $b_1 \lt \eta \lt T,$ there exists $C>0$ such that the following inequality

$\int_{0}^{T-\eta}\int_{\Omega}l^2(x,0,t)dxdt\leq C\int_{\Xi'}l^2(x,a,t)dxdadt (2.8)$

holds.

First we recall the observability inequality for the parabolics equations:
Proposition 2.2
Let $T>0,$ $t_0$ and $t_1$ such that $0 \lt t_0 \lt t_1 \lt T.$ Therefore, for all $w_0\in L^2(\Omega),$ the solution $w$ of the system:

$\left\lbrace \begin{array}{ll} \dfrac{\partial w(x,\lambda)}{\partial \lambda}-K_m\Delta w(x,\lambda)=0&\text{ in } (t_0,T)\times \Omega,\\ w=0 &\text{ on } (t_0,T)\times \partial\Omega, \\ w(x,t_0)=w_0(x)&\text{ in }\Omega, \end{array} \right. (2.9)$

verifies the following estimates

$\int_{\Omega}w^2(T,x)dx\leq\int_{\Omega}w^2(x,t_1)dx\leq c_1e^{\dfrac{c_2}{t_1-t_0}}\int_{t_0}^{t_1}\int_{\omega}w^2(x,\lambda)dxd\lambda, (2.10)$

where the constants $c_1$ and $c_2$ depend of $T$ and $\Omega.$

Proof of the inequality 2.2
Let $\tilde{n}(x,a,t)=n(x,a,t)e^{-\int_{0}^{a}\mu(\alpha)d\alpha}.$ Then $\tilde{n}$ satisfies

$\left\lbrace \begin{array}{ll} \dfrac{\partial \tilde{n}}{\partial t}+\dfrac{\partial \tilde{n}}{\partial a} +K_m\Delta\tilde{n}=0&\text{ in }\Omega\times (0,a_2)\times (0,T), \\ \hat{n}=0&\text{ on } \partial \Omega\times (0,a_2)\times (0,T),\\ \hat{n}(.,.,T)=n_Te^{-\int_{0}^{a}\mu_m(\alpha)d\alpha}&\text{ in }\Omega\times (0,A). \end{array} \right. (2.11)$

Proving the inequality (2.7) leads also to show that, there exits a constant $C>0$ such that the solution $\tilde{n}$ of (2.11) satisfies

$\int_{0}^{T-\eta}\int_{\Omega}\tilde{n}^2(x,0,t)dxdt\leq C\int_{0}^{T}\int_{a_1}^{a_2}\int_{\omega}\tilde{n}(x,a,t)dxdadt. (2.12)$

Let:

$w(\lambda)=\tilde{n}(x,T-\lambda,T+t-\lambda) \text{ ; }(\lambda\in (T-a_2,T)\hbox{ and } x\in \Omega).$

Then, $w$ verifies the following system:

$\left\lbrace \begin{array}{ll} \dfrac{\partial w(\lambda)}{\partial \lambda}-k_m \Delta w(\lambda)=0&\text{ in } \Omega\times (T-a_2,T),\\ w=0&\text{ on } \partial \Omega\times (T-a_2,T),\\ w(0) =\tilde{n}(x,T,T+t)& \text{ in }\Omega. \end{array} \right. (2.13)$

Using the Proposition 2.1 with $T-a_2 \lt t_0 \lt t_1 \lt T$ we obtain:

$\int_{\Omega}w^2(T)dx\leq\int_{\Omega}w^2(t_1)dx\leq c_1e^{\dfrac{c_2}{t_1-t_0}}\int_{t_0}^{t_1}\int_{\Omega}w^2(\lambda)dxd\lambda.$

That is equivalent to

$\int_{\Omega}\tilde{n}^2(x,0,t)dx\leq c_1e^{\dfrac{c_2}{t_1-t_0}}\int_{t_0}^{t_1}\int_{\Omega}\tilde{n}^2(x,T-\lambda,t+T-\lambda)dxd\lambda$

 

$C\int_{T-t_1}^{T-t_0}\int_{\Omega}\tilde{n}^2(x,a,t+a)dxda.$

By adequately subdividing $(0,T-\eta)$ and making judicious choices of $t_0$ and $t_1$, we obtain the result. $\square$

Figure 4. Estimation of $n(x,0,t)$ and $l(x,0,t).$ Here we have chosen $a_1=b_1$ and $a_2=b_2=b.$.Since $t\in (0,T-a_1)$ all the backward characteristics starting from $(0,t)$ enter the observation domain

We state these two propositions necessary for the proof of the inequality:
Proposition 2.3
Under the assumptions $(H_1)-(H3),$ for all $T>\max\{a_1,A-a_2\}$, there exists $C_T>0$ such that the solution $(n,l)$ of the system (2.4) verifies the following inequality:

$\int_{0}^{A}\int_{\Omega}n^{2}(x,a,0)dxda\leq C_T\int_{\Xi}n^{2}(x,a,t)dxdadt. (2.14)$

 

Remark 2.2

Note that here we first show that $n(x,a,0)=0$ in $(a_0,A)$ and we use the same technique as in the Proposition 2 to estimate $n(x,a,0)$ in $(0,a_0)$.

Proposition 2.4
Under the assumptions $(H_1)-(H_2)$ and the hypothesis
$b_1 \lt a_0 \lt b$ and $T \gt b_1.$ There exists $C_T>0$ such that the solution $(n,l)$ of the system (2.4) verifies the following inequality:

$\int_{0}^{a_0}\int_{\Omega}l^{2}(x,a,0)dxda\leq C_T\int_{\Xi'}l^{2}(x,a,t)dxdadt. (2.15)$

Let us gives a preliminary results for the proof of the Theorem 2.2 of [1].

Let $l=u_1+u_2$ where $u_1$ and $u_2$ verify

$\left\lbrace \begin{array}{ll} -\dfrac{\partial u_1}{\partial t}-\dfrac{\partial u_1}{\partial a}-K_f\Delta u_1+\mu_f(a)u_1=0&\hbox{ in }Q_{A}\times (0,T-\eta) ,\\ u_1(\sigma,a,t)=0&\hbox{ on }\partial\Omega\times (0,A)\times (0,T-\eta),,\\ u_1\left( x,A,t\right) =0&\hbox{ in } \Omega\times (0,T-\eta) \\ u_1\left(x,a,T-\eta\right)=l_{\eta}&\hbox{ in }Q_{A}. \end{array}\right. (2.16)$

and

$\left\lbrace \begin{array}{ll} -\dfrac{\partial u_2}{\partial t}-\dfrac{\partial u_2}{\partial a}-K_f\Delta u_2+\mu_f(a)u_2=V(x,a,t)&\hbox{ in }Q_{A}\times (0,T-\eta) ,\\ u_2(\sigma,a,t)=0&\hbox{ on }\partial\Omega\times (0,A)\times (0,T-\eta),\\ u_2\left( x,A,t\right) =0&\hbox{ in } \Omega\times (0,T-\eta) \\ u_2\left(x,a,T-\eta\right)=0&\hbox{ in }Q_{A}. \end{array}\right. (2.17)$

where the couple $(n,l)$ verifies

$\left\lbrace \begin{array}{ll} -n_t-n_a-K_m\Delta n+\mu_m n=0 &\text{ in }Q ,\\ -l_t-l_a-K_f\Delta l+\mu_f l=V(x,a,t)&\text{ in }Q,\\ n(\sigma,a,t)=l(\sigma,a,t)=0&\text{ on }\Sigma, \\ n(x,a,T)=n_T\quad l(x,a,T)=l_T&\text{ in }Q_A,\\ n(x,A,t)=0,\quad l(x,A,t)=0& \text{ in }Q_T.\\ \end{array} \right. (2.18)$

(see for instance [1]), with $l_{\eta}=l(x,a,T-\eta)$ in $Q_{A}$ and

$V(x,a,t)=\beta(a,p)l(x,0,t)+\beta(a,p)n(x,0,t).$

Using Duhamel’s formula we can write

$u_2(x,a,t)=\int\limits_{t}^{T-\eta}\mathbb{T}_{t-f}V(x,a,f)df$

where $\mathbb{T}$ is the semigroup generated by the operator $-K_f\Delta-\dfrac{\partial }{\partial a}+\mu_f(a).$

Proof. Proof of the Theorem 2.2
We split the term to be estimated as follows

$\int\limits_{0}^{A}\int_{\Omega}l^2(x,a,0)dxda=\int\limits_{0}^{\max\{a_1,b_1\}}\int_{\Omega}l^2(x,a,0)dxda+\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}l^2(x,a,0)dxda.$

As, $\max\{a_1,b_1\} \lt b,$ using the Proposition 2.4 of [1] we obtain the estimate

$\int\limits_{0}^{\max\{a_1,b_1\}}\int_{\Omega}l^2(x,a,0)dxda\leq C\int\limits_{0}^{T}\int\limits_{b_1}^{b_2}\int_{\omega}l^2(x,a,t)dxdadt. (2.19)$

We are now left with the estimation of

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}l^2(x,a,0)dxda.$

But since $l=u_1+u_2$, we must therefore estimate

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{1}^{2}(x,a,0)dxda+\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{2}^{2}(x,a,0)dxda.$

We have

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{2}^{2}(x,a,0)dxda\leq C_{\eta,T}(A-\max\{a_1,b_1\})\left(\int\limits_{0}^{T-\eta}\int_{\Omega}l^2(x,0,t)dxdt+\int\limits_{0}^{T-\eta}\int_{\Omega}n^2(x,0,t)dxdt\right).$

And then, using the Proposition 2.1 of [1], with $\eta=\max\{a_1,b_1\}+\delta<T\hbox{, }\delta>0,$ we obtain

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{2}^{2}(x,a,0)dxda\leq C_{\eta,T}\left(\int\limits_{0}^{T}\int\limits_{b_1}^{b_2}\int_{\omega}l^2(x,a,t)dxdadt+\int\limits_{0}^{T}\int\limits_{a_1}^{a_2}\int_{\omega}n^2(x,a,t)dxdadt\right). (2.20)$

As,

$T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\},$

we can choose $\delta>0$ small enough ($\delta$ should also check, $\max\{a_1,b_1\} \lt \min\{a_2,b_2\}-\delta$)
such that

$T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\}+2\delta;$

therefore

$T-(\max\{a_1,b_1\}+\delta)>A-(\min\{a_2,b_2\}-\delta).$

Moreover for

$a\in (\min\{a_2,b_2\}-\delta,A)$

we have

$T-(\max\{a_1,b_1\}+\delta)>A-(\min\{a_2,b_2\}-\delta)>A-a.$

Then, from the Remark 2.2 of [1]

$u_1(x,a,T)=0\hbox{ a.e. }x\in \Omega\hbox{, }a\in(\min\{a_2,b_2\}-\delta),A).$

Therefore

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{1}^2(x,a,0)dxda=\int\limits_{\max\{a_1,b_1\}}^{\min\{a_2,b_2\}-\delta}\int_{\Omega}u_{1}^{2}(x,a,0)dxda.$

As

$T-(\max\{a_1,b_1\}+\delta)>A-(\min\{a_2,b_2\}-\delta)),$

then using the Proposition 2.3 of [1], we obtain

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{1}^{2}(x,a,0)dxda\leq K\int\limits_{0}^{T-\eta}\int\limits_{b_1}^{b_2}\int_{\omega}u_{1}^{2}(x,a,t)dxdadt. (2.21)$

As

$u_1=l-u_2,$

then

$\int\limits_{0}^{T-\eta}\int\limits_{b_1}^{b_2}\int_{\omega}u_{1}^{2}(x,a,t)dxdadt$

$\leq 2\left(\int\limits_{0}^{T-\eta}\int\limits_{b_1}^{b_2}\int_{\omega}u_{2}^{2}dxdadt+\int\limits_{0}^{T-\eta}\int\limits_{b_1}^{b_2}\int_{\omega}l^{2}(x,a,t)dxdadt\right). (2.22)$

Moreover, under the assumption of Theorem 1.1 of [1], the solution $u_2$ of the system (2.22) verifies the following estimate :

$\int\limits_{0}^{T-\eta}\int\limits_{b_1}^{b_2}\int_{\omega}u_{2}^2(x,a,t)dxdadt\leq \int\limits_{0}^{T-\eta}\int\limits_{0}^{A}\int_{\Omega}u_{2}^2(x,a,t)dxdadt$

$\leq C \left(\int\limits_{0}^{T-\eta}\int_{\Omega}l^2(x,0,t)dxdt+\int\limits_{0}^{T-\eta}\int_{\Omega}n^2(x,0,t)dxdt\right). (2.23)$

where $C=e^{\frac{3}{2}(T-\eta)}\|\beta\|^{2}_{\infty}A.$

From the Proposition 2.1 of [1], we get

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}u_{1}^{2}(x,a,0)dxda\leq C(T,\eta,\|\beta\|_{\infty})\left(\int\limits_{0}^{T}\int\limits_{b_1}^{b_2}\int_{\omega}l^{2}(x,a,t)dxdadt+\int\limits_{0}^{T}\int\limits_{a_1}^{a_2}\int_{\omega}n^{2}(x,a,t)dxdadt\right). (2.24)$

Combining the inequalities (2.23) and (2.24), we obtain

$\int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}l^{2}(x,a,0)dxda\leq C(T,\eta,\|\beta\|_{\infty}) \left(\int\limits_{0}^{T}\int\limits_{a_1}^{a_2}\int_{\omega}n^{2}(x,a,t)dxdadt+\int\limits_{0}^{T}\int\limits_{b_1}^{b_2}\int_{\omega}n^{2}(x,a,t)dxdadt\right). (2.25)$

Therefore, (2.24) and (2.25) give

$\int\limits_{0}^{A}\int_{\Omega}l^{2}(x,a,0)dxda\leq K_{T} \left(\int\limits_{0}^{T}\int\limits_{a_1}^{a_2}\int_{\omega}n^{2}(x,a,t)dxdadt+\int\limits_{0}^{T}\int\limits_{b_1}^{b_2}\int_{\omega}n^{2}(x,a,t)dxdadt\right). (2.26)$

Finally, combining (2.26) and the inequality of the Proposition 2.3 of [1], we get the observability inequality.

Illustration of the observability inequality and estimation of the non local terms
Let $\Lambda$ be a operator define as follow:

$\Lambda: L^{2}(Q_T)\longrightarrow L^{2}(Q_T)\quad p\longmapsto \int_{0}^{A}\lambda (a)m(p)da (2.27)$

where the couple $(m(p),f(p))$ is the solution of the following auxilliary system verifying

$m(x,a,T)=0 \hbox{ a.e. } x\in \Omega \hbox{ } a\in (0,A), (2.28)$

$f(x,a,T)=0 \hbox{ a.e. } x\in \Omega\hbox{ } a\in (0,A). (2.29)$

Under the assumptions of Theorem 1.1, we can show that the operator $\Lambda$ is continuous, and the set $\Lambda(L^{2}(Q_T))$ is relatively compact in $L^{2}(Q_T)$.
By Schauder’s fixed point theorem $\Lambda$ admits a fixed point and we get the reult of the Theorem 1.1. Let us gives a preliminary results for the proof of the Theorem 3.1. of [1].
We denote by

$Q=\Omega\times (0,A)\times (0,T)\hbox{, }Q_A=\Omega\times(0,A)\hbox{, }Q_T=\Omega\times (0,T)\hbox{ and }\Sigma=\partial\Omega\times (0,A)\times (0,T). \square$

 

Numerics Simulations without the space variable

We start with a mesh of segment $[0,A]$. By the method of finite differences we get,

$\dfrac{\partial m}{\partial a}(a_j,t)=\dfrac{m(a_{j},t)-m(a_{j-1},t)}{\Delta a},$

$\dfrac{\partial f}{\partial a}(a_j,t)=\dfrac{f(a_{j},t)-f(a_{j-1},t)}{\Delta a},$

which corresponds to the approximation of the aging terms at the different points of the mesh. Let $m_{j}(t)$ and $f_{j}(t)$ the approximation of $m(a_j,t)$, $f(a_j,t)$ and $U_{l}(t) = (m_{j}(t),f_{i}(t))_{1\leq j\leq N ,\quad 1\leq i\leq N}$, and $Y_l=(y(a_i))_{1\leq i\leq N}$ the initial condition vector and $A_l$ the matrix of the mortality approximation and the aging approximation.
So we get the following system

$\dot{U_l}=A_lU_l+\text{vectorF}+BV_l\text{ with }U_l(0)=Y_l$

where $B$ is a matrix defining the control support of the matrices, vectorF is the contribution of the nonlocal part, which comes from the births and $V_l(t)$ the control vector which also comes from the nonlocal term.
Using matlab’s ODE 45 we obtain the following results:

Figure 5. Solution of the male

Figure 6. Solution of the female.

For the approximate null controllability we minimize the functional

$J_\epsilon(V_l)=\dfrac{1}{\epsilon}\int\limits_{0}^{A}|U_l(T)|^2+\int\limits_{0}^{T}\int\limits_{0}^{A}|V_l(t)|^2dadt$

under the constraint

$\dot{U_l}=A_lU_l+\text{vectorF}+BV_l\text{ et }U_l(0)=(m_{00},f_{00}).$

Using Casadi’s algorithm, we can see the following result:

Figure 7. Solution of the male

Figure 8. Solution of the female.

 

References

[1] Y. Simporé, O. Traoré. Null controllability of a nonlinear age, space and two-sex structured population dynamics model. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021052