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

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 Ξ=ω×(a1,a2)×(0,T)Q\Xi=\omega\times(a_1,a_2)\times (0,T)\subset Q and Ξ=ω×(b1,b2)×(0,T)Q\Xi'=\omega'\times(b_1,b_2)\times (0,T)\subset Q where Q=Ω×(0,A)×(0,T).Q=\Omega\times(0,A)\times(0,T). We denote also Σ=Ω×(0,A)×(0,T)QT=Ω×(0,T)\Sigma=\partial\Omega\times(0,A)\times(0,T)\hbox{, }Q_T=\Omega\times(0,T) and QA=Ω×(0,A).Q_A=\Omega\times(0,A).

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

{mt+maKmΔm+μmm=χΞvm in Q,ft+faKfΔf+μff=χΞvf in Q,m(σ,a,t)=f(σ,a,t)=0 on Σ,m(x,a,0)=m0f(x,a,0)=f0 in QA,m(x,0,t)=(1γ)N(x,t)f(x,0,t)=γN(x,t) in QT,N(x,t)=0Aβ(a,M)fdaM=0Aλ(a)mda in QT.(1.1) \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 m0L2(QA)m_0\in L^2(Q_A), f0L2(QA)f_0\in L^2(Q_A), vmL2(Q)v_m\in L^2(Q), vfL2(Q)v_f\in L^2(Q) and γ(0,1).\gamma\in(0,1).

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

(H1):{(i) μm0,μf0 a.e. in [0,A], (ii) μmLloc1([0,A)),μfLloc1([0,A)),(iii) 0Aμm(a)da=+,0Aμf(a)da=+. (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
Πm(a)=e0aμm(s)ds and Πf(a)=e0aμf(s)ds\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 aa and female individuals of age a,a, respectively.

(H2):{(i) βC([0,A]×R),(ii) β(a,p)0 for all (a,p)[0,A]×R,(iii) β(a,0)=0 in (0,A). (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.

 

(H3):{λC1([0,A]),λ0 for all a[0,A]. (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:

(H4):{(i) there exists b(0,A) such that β(a,p)=0,(a,p)[0,b)×R,(ii)  there exists L>0 such that β(a,p)β(a,q)Lpq for all p,qRa[0,A],(iii) there exists β0>0 such that 0β(a,p)β0(a,p)[0,A]×R.(1.2) (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)

 

(H5):{λμmL1((0,A)). (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 (H1)(H2)(H3)(H4)(H5)(H_1)-(H_2)-(H_3)-(H_4)-(H_5) hold. If (0,b)(a1,a2)(b1,b2),(0,b)\cap(a_1,a_2)\cap(b_1,b_2)\neq \emptyset, for every time T>max{a1,b1}+max{Aa2,Ab2}T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\} and for every (m0,f0)(L2(QA))2,(m_0,f_0)\in \left(L^2(Q_A)\right)^2, there exists (vm,vf)L2(Ξ)×L2(Ξ)(v_m,v_f)\in L^2(\Xi)\times L^2(\Xi') such the solution (m,f)(m,f) of the system (1.1) verifies:

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

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

Remark 2.1.

Notice that ωω\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 a1a_1 and a2a_2 (t3t1=Aa2+ϵ+a1t_3-t_1=A-a_2+\epsilon+a_1).In this case, all males and females whose age is greater than a2a_2 can live to the maximum age AA and can have offspring which can also live to the age of a1a_1.

2.2 Null controllability of auxiliary system

Let pp be a function in L2(QT),L^{2}(Q_T), we define the auxilliary system given by:

{mt+maKmΔm+μmm=χΞv in Q,ft+faKfΔf+μff=χΞu in Q,m(σ,a,t)=f(σ,a,t)=0 on Σ,m(x,a,0)=m0f(x,a,0)=f0 in QA,m(x,0,t)=(1γ)0Aβ(a,p)fda,f(x,0,t)=γ0Aβ(a,p)fda in QT.(2.3) \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)(L2((0,A)×(0,T);H01(Ω)))2(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{a1,b1}+max{Aa2,Ab2}.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)(n,l) be the solution of the following adjoint system to the auxilliary system (2.3)

{ntnaKmΔn+μmn=0 in Q,ltlaKfΔl+μfl=(1γ)β(a,p)n(x,0,t)+γβ(a,p)l(x,0,t) in Q,n(σ,a,t)=l(σ,a,t)=0 on Σ,n(x,a,T)=nTl(x,a,T)=lT in QA,n(x,A,t)=0,l(x,A,t)=0 in QT.(2.4) \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,T, we have the following:

2.3 Observability Inequality

Theorem 2.2

Under the assumptions of Theorem 1, for every T>max{a1,b1}+max{Aa2,Ab2},T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\}, there exists a constant CT>0C_{T}>0 independent of pp such that the solution (n,l)(n,l) of the system (2.4) verifies:

0AΩn2(x,a,0)dxda+0AΩl2(x,a,0)dxda \int_{0}^{A}\int_{\Omega}n^2(x,a,0)dxda+\int_{0}^{A}\int_{\Omega}l^2(x,a,0)dxda

CT(Ξn2(x,a,t)dxdadt+Ξl2(x,a,t)dxdadt). \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 (nT,lT)(L2(QA))2,(n_T,l_T)\in(L^2(Q_A))^2, under the assumptions (H1)(H_1) and (H2),(H_2), the cascade system (2.4) admits a unique solution (n,l).(n,l). Moreover, integrating along the characteristics line the solution (n,l)(n,l) of (2.4) is given by:

n(t)={π1(a+Tt)π1(a)e(Tt)KmΔnT(x,a+Tt) if TtAa,0 if Aa<Tt,(2.5) 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)={π2(a+Tt)π2(a)e(Tt)KfΔlT(x,a+tT)+tTπ2(a+st)π2(a)e(st)KfΔβ(a+st,p(x,s))((1γ)n(x,0,s)+γl(x,0,s))ds in D1,tt+Aaπ2(a+st)π2(a)e(st)KfΔβ(a+st,p(x,s))((1γ)n(x,0,s)+γl(x,0,s))ds in D2,(2.6) 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 π1(a)=e0aμm(r)drπ2(a)=e0aμf(r)dretKmΔ\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 KmΔ-K_m\Delta with the Dirichlet boundary condition and

D1={(a,t)(0,A)×(0,T) such that TtAa}, D_1=\{(a,t)\in (0,A)\times (0,T) \text{ such that } T-t\leq A-a\},

D2={(a,t)(0,A)×(0,T) such that Tt>Aa}. D_2=\{(a,t)\in (0,A)\times (0,T) \text{ such that } T-t > A-a\}.

Using the fact that β(a,p)=0 for all a[0,b).\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 η\eta satisfying a1<η<T,a_1 \lt \eta \lt T, there exists C>0C>0 such that the following inequality

0TηΩn2(x,0,t)dxdtC0Ta1a2ωn2(x,a,t)dxdadt(2.7) \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 η\eta verifying b1<η<T,b_1 \lt \eta \lt T, there exists C>0C>0 such that the following inequality

0TηΩl2(x,0,t)dxdtCΞl2(x,a,t)dxdadt(2.8) \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, t0t_0 and t1t_1 such that 0<t0<t1<T.0 \lt t_0 \lt t_1 \lt T. Therefore, for all w0L2(Ω),w_0\in L^2(\Omega), the solution ww of the system:

{w(x,λ)λKmΔw(x,λ)=0 in (t0,T)×Ω,w=0 on (t0,T)×Ω,w(x,t0)=w0(x) in Ω,(2.9) \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

Ωw2(T,x)dxΩw2(x,t1)dxc1ec2t1t0t0t1ωw2(x,λ)dxdλ,(2.10) \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 c1c_1 and c2c_2 depend of TT and Ω.\Omega.

Proof of the inequality 2.2
Let n~(x,a,t)=n(x,a,t)e0aμ(α)dα.\tilde{n}(x,a,t)=n(x,a,t)e^{-\int_{0}^{a}\mu(\alpha)d\alpha}. Then n~\tilde{n} satisfies

{n~t+n~a+KmΔn~=0 in Ω×(0,a2)×(0,T),n^=0 on Ω×(0,a2)×(0,T),n^(.,.,T)=nTe0aμm(α)dα in Ω×(0,A).(2.11) \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>0C>0 such that the solution n~\tilde{n} of (2.11) satisfies

0TηΩn~2(x,0,t)dxdtC0Ta1a2ωn~(x,a,t)dxdadt.(2.12) \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(λ)=n~(x,Tλ,T+tλ) ; (λ(Ta2,T) and xΩ). w(\lambda)=\tilde{n}(x,T-\lambda,T+t-\lambda) \text{ ; }(\lambda\in (T-a_2,T)\hbox{ and } x\in \Omega).

Then, ww verifies the following system:

{w(λ)λkmΔw(λ)=0 in Ω×(Ta2,T),w=0 on Ω×(Ta2,T),w(0)=n~(x,T,T+t) in Ω.(2.13) \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 Ta2<t0<t1<TT-a_2 \lt t_0 \lt t_1 \lt T we obtain:

Ωw2(T)dxΩw2(t1)dxc1ec2t1t0t0t1Ωw2(λ)dxdλ. \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

Ωn~2(x,0,t)dxc1ec2t1t0t0t1Ωn~2(x,Tλ,t+Tλ)dxdλ \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

 

CTt1Tt0Ωn~2(x,a,t+a)dxda. C\int_{T-t_1}^{T-t_0}\int_{\Omega}\tilde{n}^2(x,a,t+a)dxda.

By adequately subdividing (0,Tη)(0,T-\eta) and making judicious choices of t0t_0 and t1t_1, we obtain the result. \square


Figure 4. Estimation of n(x,0,t)n(x,0,t) and l(x,0,t).l(x,0,t). Here we have chosen a1=b1a_1=b_1 and a2=b2=b.a_2=b_2=b..Since t(0,Ta1)t\in (0,T-a_1) all the backward characteristics starting from (0,t)(0,t) enter the observation domain

We state these two propositions necessary for the proof of the inequality:
Proposition 2.3
Under the assumptions (H1)(H3),(H_1)-(H3), for all T>max{a1,Aa2}T>\max\{a_1,A-a_2\}, there exists CT>0C_T>0 such that the solution (n,l)(n,l) of the system (2.4) verifies the following inequality:

0AΩn2(x,a,0)dxdaCTΞn2(x,a,t)dxdadt.(2.14) \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)=0n(x,a,0)=0 in (a0,A)(a_0,A) and we use the same technique as in the Proposition 2 to estimate n(x,a,0)n(x,a,0) in (0,a0)(0,a_0).

Proposition 2.4
Under the assumptions (H1)(H2)(H_1)-(H_2) and the hypothesis
b1<a0<bb_1 \lt a_0 \lt b and T>b1.T \gt b_1. There exists CT>0C_T>0 such that the solution (n,l)(n,l) of the system (2.4) verifies the following inequality:

0a0Ωl2(x,a,0)dxdaCTΞl2(x,a,t)dxdadt.(2.15) \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=u1+u2l=u_1+u_2 where u1u_1 and u2u_2 verify

{u1tu1aKfΔu1+μf(a)u1=0 in QA×(0,Tη),u1(σ,a,t)=0 on Ω×(0,A)×(0,Tη),,u1(x,A,t)=0 in Ω×(0,Tη)u1(x,a,Tη)=lη in QA.(2.16) \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

{u2tu2aKfΔu2+μf(a)u2=V(x,a,t) in QA×(0,Tη),u2(σ,a,t)=0 on Ω×(0,A)×(0,Tη),u2(x,A,t)=0 in Ω×(0,Tη)u2(x,a,Tη)=0 in QA.(2.17) \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)(n,l) verifies

{ntnaKmΔn+μmn=0 in Q,ltlaKfΔl+μfl=V(x,a,t) in Q,n(σ,a,t)=l(σ,a,t)=0 on Σ,n(x,a,T)=nTl(x,a,T)=lT in QA,n(x,A,t)=0,l(x,A,t)=0 in QT.(2.18) \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 lη=l(x,a,Tη)l_{\eta}=l(x,a,T-\eta) in QAQ_{A} and

V(x,a,t)=β(a,p)l(x,0,t)+β(a,p)n(x,0,t). 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

u2(x,a,t)=tTηTtfV(x,a,f)df u_2(x,a,t)=\int\limits_{t}^{T-\eta}\mathbb{T}_{t-f}V(x,a,f)df

where T\mathbb{T} is the semigroup generated by the operator KfΔa+μf(a).-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

0AΩl2(x,a,0)dxda=0max{a1,b1}Ωl2(x,a,0)dxda+max{a1,b1}AΩl2(x,a,0)dxda. \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{a1,b1}<b,\max\{a_1,b_1\} \lt b, using the Proposition 2.4 of [1] we obtain the estimate

0max{a1,b1}Ωl2(x,a,0)dxdaC0Tb1b2ωl2(x,a,t)dxdadt.(2.19) \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

max{a1,b1}AΩl2(x,a,0)dxda. \int\limits_{\max\{a_1,b_1\}}^{A}\int_{\Omega}l^2(x,a,0)dxda.

But since l=u1+u2l=u_1+u_2, we must therefore estimate

max{a1,b1}AΩu12(x,a,0)dxda+max{a1,b1}AΩu22(x,a,0)dxda. \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

max{a1,b1}AΩu22(x,a,0)dxdaCη,T(Amax{a1,b1})(0TηΩl2(x,0,t)dxdt+0TηΩn2(x,0,t)dxdt). \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 η=max{a1,b1}+δ<Tδ>0,\eta=\max\{a_1,b_1\}+\delta<T\hbox{, }\delta>0, we obtain

max{a1,b1}AΩu22(x,a,0)dxdaCη,T(0Tb1b2ωl2(x,a,t)dxdadt+0Ta1a2ωn2(x,a,t)dxdadt).(2.20) \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{a1,b1}+max{Aa2,Ab2}, T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\},

we can choose δ>0\delta>0 small enough (δ\delta should also check, max{a1,b1}<min{a2,b2}δ\max\{a_1,b_1\} \lt \min\{a_2,b_2\}-\delta)
such that

T>max{a1,b1}+max{Aa2,Ab2}+2δ; T>\max\{a_1,b_1\}+\max\{A-a_2,A-b_2\}+2\delta;

therefore

T(max{a1,b1}+δ)>A(min{a2,b2}δ). T-(\max\{a_1,b_1\}+\delta)>A-(\min\{a_2,b_2\}-\delta).

Moreover for

a(min{a2,b2}δ,A) a\in (\min\{a_2,b_2\}-\delta,A)

we have

T(max{a1,b1}+δ)>A(min{a2,b2}δ)>Aa. T-(\max\{a_1,b_1\}+\delta)>A-(\min\{a_2,b_2\}-\delta)>A-a.

Then, from the Remark 2.2 of [1]

u1(x,a,T)=0 a.e. xΩa(min{a2,b2}δ),A). u_1(x,a,T)=0\hbox{ a.e. }x\in \Omega\hbox{, }a\in(\min\{a_2,b_2\}-\delta),A).

Therefore

max{a1,b1}AΩu12(x,a,0)dxda=max{a1,b1}min{a2,b2}δΩu12(x,a,0)dxda. \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{a1,b1}+δ)>A(min{a2,b2}δ)), T-(\max\{a_1,b_1\}+\delta)>A-(\min\{a_2,b_2\}-\delta)),

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

max{a1,b1}AΩu12(x,a,0)dxdaK0Tηb1b2ωu12(x,a,t)dxdadt.(2.21) \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

u1=lu2, u_1=l-u_2,

then

0Tηb1b2ωu12(x,a,t)dxdadt \int\limits_{0}^{T-\eta}\int\limits_{b_1}^{b_2}\int_{\omega}u_{1}^{2}(x,a,t)dxdadt

2(0Tηb1b2ωu22dxdadt+0Tηb1b2ωl2(x,a,t)dxdadt).(2.22) \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 u2u_2 of the system (2.22) verifies the following estimate :

0Tηb1b2ωu22(x,a,t)dxdadt0Tη0AΩu22(x,a,t)dxdadt \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

C(0TηΩl2(x,0,t)dxdt+0TηΩn2(x,0,t)dxdt).(2.23) \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=e32(Tη)β2A.C=e^{\frac{3}{2}(T-\eta)}\|\beta\|^{2}_{\infty}A.

From the Proposition 2.1 of [1], we get

max{a1,b1}AΩu12(x,a,0)dxdaC(T,η,β)(0Tb1b2ωl2(x,a,t)dxdadt+0Ta1a2ωn2(x,a,t)dxdadt).(2.24) \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

max{a1,b1}AΩl2(x,a,0)dxdaC(T,η,β)(0Ta1a2ωn2(x,a,t)dxdadt+0Tb1b2ωn2(x,a,t)dxdadt).(2.25) \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

0AΩl2(x,a,0)dxdaKT(0Ta1a2ωn2(x,a,t)dxdadt+0Tb1b2ωn2(x,a,t)dxdadt).(2.26) \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 Λ\Lambda be a operator define as follow:

Λ:L2(QT)L2(QT)p0Aλ(a)m(p)da(2.27) \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))(m(p),f(p)) is the solution of the following auxilliary system verifying

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

f(x,a,T)=0 a.e. xΩ a(0,A).(2.29)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 Λ\Lambda is continuous, and the set Λ(L2(QT))\Lambda(L^{2}(Q_T)) is relatively compact in L2(QT)L^{2}(Q_T).
By Schauder’s fixed point theorem Λ\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=Ω×(0,A)×(0,T)QA=Ω×(0,A)QT=Ω×(0,T) and Σ=Ω×(0,A)×(0,T). 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][0,A]. By the method of finite differences we get,

ma(aj,t)=m(aj,t)m(aj1,t)Δa, \dfrac{\partial m}{\partial a}(a_j,t)=\dfrac{m(a_{j},t)-m(a_{j-1},t)}{\Delta a},

fa(aj,t)=f(aj,t)f(aj1,t)Δa, \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 mj(t)m_{j}(t) and fj(t)f_{j}(t) the approximation of m(aj,t)m(a_j,t), f(aj,t)f(a_j,t) and Ul(t)=(mj(t),fi(t))1jN,1iNU_{l}(t) = (m_{j}(t),f_{i}(t))_{1\leq j\leq N ,\quad 1\leq i\leq N}, and Yl=(y(ai))1iNY_l=(y(a_i))_{1\leq i\leq N} the initial condition vector and AlA_l the matrix of the mortality approximation and the aging approximation.
So we get the following system

Ul˙=AlUl+vectorF+BVl with Ul(0)=Yl \dot{U_l}=A_lU_l+\text{vectorF}+BV_l\text{ with }U_l(0)=Y_l

where BB is a matrix defining the control support of the matrices, vectorF is the contribution of the nonlocal part, which comes from the births and Vl(t)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

Jϵ(Vl)=1ϵ0AUl(T)2+0T0AVl(t)2dadt 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

Ul˙=AlUl+vectorF+BVl et Ul(0)=(m00,f00). \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