Null Controllability of a Nonlinear Age Structured Model for a Two-Sex Population

This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at the birth level. In this work, we consider two cases of null controllability problem. The first problem is related to the extinction of male and female subpopulation density. The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if 
 
 A
 
 is the maximal age, a time interval of duration 
 
 A
 
 after the extinction of males or females, one must get the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after Kakutani’s fixed-point theorem.


Introduction
In practice, this study takes place in the fight against malaria. Malaria is a serious disease (in 2017, there were 219 million cases in the world [1]), and our work takes its importance in the strategy to fight against it.
In fact, malaria is a vector-borne disease transmitted by an infective female anopheles mosquito. A malaria control strategy in Brazil or Burkina Faso consists of releasing genetically modified male mosquitoes (precisely sterile males) in the nature. This can reduce the reproduction of mosquitoes since females mate only once in their life cycle.
In the theoretical framework, very few authors have studied control problems of a two-sex structured population dynamics model.
The control problems of coupled systems of population dynamics models take an intense interest and are widely investigated in many papers. Among them, we have [2][3][4] and the references therein. In fact, in [2], the authors studied a coupled reaction-diffusion equations describing interaction between a prey population and a predator one. The goal of the above work is to look for a suitable control supported on a small spatial subdomain which guarantees the stabilization of the predator population to zero. In [4], the objective was different. More precisely, the authors consider an agedependent prey-predator system and they prove the existence and uniqueness of an optimal control (called also "optimal effort") which gives the maximal harvest via the study of the optimal harvesting problem associated with their coupled model. In [5], He and Ainseba study the null controllability of a butterfly population by acting on eggs, larvas, and female moths in a small age interval.
In [3], the authors analyze the growth of a two-sex population with a fixed age-specific sex ratio without diffusion. The model is intended to give an insight into the dynamics of a population where the mating process takes place at random choice and the proportion between females and males is not influenced by environmental or social factors but only depends on a differential mortality or on a possible transition from one sex to the other (e.g., in sequential hermaphrodite species). Simpore and Traore study in [6] the null controllability of a nonlinear age, space, and two-sex structured population dynamics model. They first study an approximate null controllability result for an auxiliary cascade system and prove the null controllability of the nonlinear system by means of Schauder's fixed-point theorem.
Unlike the model treated in [6], we consider a nonlinear cascade system with two different fertility rates and without space variable. The fertility rate of the male λ and the fertility rate β of the female depend on the total population of the fertile males.
Remark 1. Here, we are talking about the null controllability as an extinction of the population, but we can see in the sense of exact controllability to the trajectories since there is an equivalence between the null controllability and exact controllability to the trajectories in the linear case.

Model and Main Results
In this paper, we study the null controllability of an infinite dimensional nonlinear coupled system describing the dynamics of a two-sex structured population. Let ðm, f Þ be the solution of the following system: where T is a positive number, Q = ð0, AÞ × ð0, TÞ, Θ = ð0, a 2 Þ × ð0, TÞ, Θ 1 = ða 1 , a 2 Þ × ð0, TÞ, and Θ 2 = ðb 1 , b 2 Þ × ð0, TÞ: Here, 0 ≤ a 1 < a 2 ≤ A, 0 ≤ b 1 < b 2 ≤ A, Q A = ð0, AÞ, and Q T = ð0, TÞ: We denote the density of males and females of age a at time t, respectively, by mða, tÞ and f ða, tÞ. Moreover, μ m and μ f denote, respectively, the natural mortality rate of males and females. The control functions are v m and v f and depend on a and t: In addition, χ Θ 1 and χ Θ 2 are the characteristic functions of the support of the control v m and v f , respectively.
We have denoted by β the positive function describing the fertility rate that depends on a and also on where λ is the fertility function of the male individuals. Thus, the densities of newborn male and female individuals at time t are given, respectively, by mð0, tÞ = ð1 − γÞNðtÞ and f ð0, tÞ = γNðtÞ where We assume that the fertility rate β, λ and the mortality rate μ f , μ m satisfy the demographic properties: We further assume that the birth function β and the fertility function λ verify the following hypothesis: The assumption βða, 0Þ = 0 for a ∈ ð0, AÞ means that the birth rate is zero if there are no fertile male individuals.
We can now state the main results. If ða 1 , a 2 Þ ⊂ ðb 1 , b 2 Þ, we have the following theorem: Theorem 2. Let us assume that the assumptions ðH 1 Þ − ðH 4 Þ hold true. If a 1 < b, for every time T > a 1 + A − a 2 and for every such that the associated solution ðm, f Þ of system (1)  We use the technique of [6,7] combining final-state observability estimates with the use of characteristics to establish the observability inequalities necessary for the null controllability property of the auxiliary systems. Roughly, in our method, we first study the null controllability result for an auxiliary cascade system. Afterwards, we prove the null controllability result for system (1) by means of Kakutani's fixed-point theorem.
The remainder of this paper is as follows: in Section 2, we describe the model and give the main results. Then, we study the existence and uniqueness of a positive solution for the model in Section 3. Section 4 is devoted to the proof of Theorem 2 and Theorem 3. Some illustrations of numerical simulations are given in Section 5.

Well-Posedness Result
In this section, we study the existence of positive solution of the model. For this, we assume that the so-called demographic conditions ðH 1 Þ, ðH 2 Þ, ðH 3 Þ, and ðH 4 Þ are verified. Moreover, here, we suppose that where K and C are positive constants.
Moreover, suppose that then, ðm, f Þ is also positive.
Proof of Theorem 5. Let p be fixed in L 2 ð0, TÞ, h and h′ be fixed in L 2 ðQÞ, and consider the following system: For every f 0 ∈ L 2 ð0, AÞ and h′ ∈ L 2 ðQÞ, the following system admits a unique positive solution in L 2 ðQÞ, see [8,9], and one has f k k 2 where C is a positive constant and independent of p because β ∈ L ∞ ðð0, TÞ × ð0, AÞÞ: Now, f and h ′ are being known; the system 3 Abstract and Applied Analysis admits a unique positive system in L 2 ðQÞ, and we have the following estimation: where K is a positive constant and independent of p because β ∈ L ∞ ðð0, TÞ × ð0, AÞÞ: Let us define Φ : L 2 + ðQÞ ⟶ L 2 + ðQÞ, ΦðpÞ = mðpÞ where mðpÞ is the unique solution of the system (15).
For any p, q ∈ L 2 + ðQÞ, we set and w = ðmðpÞ − mðqÞÞe −γ 0 t where γ 0 is a positive parameter that will be chosen later; w is a solution of Multiplying (18) by w and integrating over ð0, AÞ × ð0, tÞ, and using Young's inequality, we get Hence, for every γ 0 > 0, there is a constant C = max Now, set F = ðf ðpÞ − f ðqÞÞe −δt where δ is a positive parameter that will be chosen later. Then, F solves the following auxiliary system: Similarly as above, we have Hence, there is a positive constant C′ such that Setting YðtÞ = Ð A 0 β 1 ðaÞf ðpÞda a:e in ð0, TÞ, Y solves the following system: Abstract and Applied Analysis Multiplying (24) by Y, integrating over ð0, tÞ, and using Young's inequality, we get So, Let us setf = e −λ 0 t f : Then, from (13),f satisfies the following system: Multiplying the first equation of (27) byf , integrating on Q, and using the inequality of Young, we get Using the inequality of Cauchy-Schwarz and choosing λ 0 = ð3/2Þ + α 2 + , we obtain So, Using (26), (30), and against Young's inequality, we have From (14), we have just proved the existence of a positive constant C such that The estimate (32) means also that Y ∈ L ∞ ð0, TÞ. Combining (20), (23), and (32), we get the following estimate: where σ is a positive constant. Let us define the metric d on L 2 + ðQÞ by setting We have Abstract and Applied Analysis Using the Fubini theorem, we conclude that Then, Φ is a contraction on the complete metric space L 2 + ðQÞ into itself. Using Banach's fixed-point theorem, we conclude the existence of a unique fixed-point m. Moreover, m is nonnegative. Hence, the unique couple ðm, f Þ is the unique solution to our problem (1).

Null Controllability Results
For the sequel, the hypothesis ðH 5 Þ is not necessary. As a consequence, the uniqueness and the positivity of the solution of system (1) are not guaranteed.
We first establish an observability inequality to show the controllability of a linear system. Then, by a fixed-point method, we show the controllability of the model.

Null Controllability of an Auxiliary Coupled
System. This section is devoted to the study of an auxiliary system obtained from system (1).
Let p be a L 2 ðQ T Þ function, we define the auxiliary system given by solution ðm, f Þ ∈ ðL 2 ðQÞÞ 2 , see Section 3. The system above is null approximately controllable. Indeed, we have the following result: Theorem 6. Let us assume that assumptions ðH 1 Þ − ðH 2 Þ hold. For every time T > a 1 + A − a 2 , for every κ, θ > 0, and for every ðm 0 , f 0 Þ ∈ ðL 2 ðQ A ÞÞ 2 , there exists a control ðv κ , v θ Þ such that the solutions m and f of system (37) verify The adjoint system of (37) is given by The main idea in this part is to establish an observability inequality of (39) that will allow us to prove the approximate null controllability of (37).
The basic idea for establishing this inequality is the estimation of nonlocal terms. For that, let us start by formulating a representation of the solution of the adjoint system by the method of characteristic and semigroup.

Theorem 7.
Under the assumptions of Theorem 2, if ða 1 , a 2 Þ ⊂ ðb 1 , b 2 Þ, there exists a constant C T > 0 independent of p such that the couple ðn, lÞ solution of (39) verifies the following inequality: For the proof of Theorem 7, we state the following estimations of the nonlocal terms.

Proposition 8.
Under the assumptions of Theorem 2, there exists C > 0 such that where a 1 < η < T: Moreover, if the first condition of ðH 3 Þ hold, we have the inequality for every η such that b 1 < b and b 1 < η < T: Proof of Proposition 8. The state n of (39) verifies We denote byñða, tÞ = nða, tÞe − Ð a 0 μ m ðsÞds : Then,ñ satisfies Proving the inequality leads to getting inequality (43).

Proposition 9.
Let us assume the assumptions ðH 1 Þ − ðH 3 Þ: For every T > sup fa 1 , A − a 2 g, there exists C T > 0 such that the solution ðn, lÞ of system (37) verifies the following inequality: Note that for every T > sup fa 1 , A − a 2 g, there exists a 0 ∈ ða 1 , a 2 Þ such that nða, 0Þ = 0 for all a ∈ ða 0 , AÞ: This is a consequence of the following lemma.
Two situations can arise: (ii) a 1 < b 0 < a 0 , in this situation, we split the interval ð0, a 0 Þ as Case 2. T ≥ a 2 : In this case, we split the interval ð0, a 0 Þ as We make the proof for the second case. For the proof in the first case, see [6].
Upper bound on ð0, a 1 Þ: For a ∈ ð0, a 1 Þ, we set wðλÞ =ñðT + a − λ, T − λÞ, λ ∈ ða 1 , TÞ: We prove easily that w is a constant, see the proof of Proposition 8. Then, we set Integrating with respect to a over ð0, a 1 Þ, we get Finally, we obtain Upper bound on ða 1 , a 0 Þ: For a ∈ ða 1 , a 0 Þ, we set wðλÞ =ñðT + a − λ, T − λÞ, λ ∈ ða 0 , TÞ and Making as above and integrating with respect to a over ða 1 , a 0 Þ, it follows that The inequalities (78) and (80) give the desired result. We also need the following estimate for the proof of Theorem 7. Proposition 11. Let us assume the assumptions ðH 1 Þ − ðH 2 Þ, let b 1 < a 0 < b and T > b 1 : Then, there exists C T > 0 such that the solution l of (39) verifies Proof of Proposition 11. We suppose that β = 0 in ð0, bÞ, the function l verifies Proceeding as in the proof of Proposition 9, we get the desired result.

Abstract and Applied Analysis
For the proof of Theorem 7, we start with the following lemma.
Proof of Theorem 7. Let a 0 as in Lemma 12, we have Using the results of Lemma 12, the assumption of β, and the regularity of ðπ 2 ða + sÞÞ/ðπ 2 ðaÞÞ, we can prove the existence of a constant K T independent of p Moreover, we have b 1 ≤ a 1 ≤ a 1 + κ: Using Proposition 8, it follows that Finally, adding the above to the results of Proposition 9 and Proposition 11, we get For ε > 0 and θ > 0, we consider the functional J ε,θ defined by 10 Abstract and Applied Analysis where ðm, f Þ is the solution of the following system: Lemma 13. The functional J ε,θ is continuous, strictly convex, and coercive. Consequently, J ε,θ reaches its minimum at a point ðv m,ε , v f ,θ Þ ∈ L 2 ðΘ 1 Þ × L 2 ðΘ 2 Þ. Setting ðm ε , f θ Þ the associated solution of (93) and ðn ε , l θ Þ the solution of (39) with we have Moreover, there exist C i > 0, 1 ≤ i ≤ 4, independent of ε and θ such that Proof of Lemma 13. It is easy to check that J ε,θ is coercive, continuous, and strictly convex. Then, it admits a unique minimizer ðv ε , v θ Þ: The maximum principle gives where the couple ðn ε , l θ Þ is the solution of the system −∂ t n ε − ∂ a n ε + μ m n ε = 0 in Q, Multiplying the first and the second equation of (98) by, respectively, m ε and f θ , integrating with respect to Q and using (97), we get Combining (99) and (100), we obtain Using the inequality of Young, we have, for any δ > 0, 11 Abstract and Applied Analysis Using observability inequality (42) and choosing δ = C T in the previous inequality, it follows that This gives the desired result necessary to the proof of the main one. Now, we consider the system where ðn ε , l θ Þ is the solution of (98) that minimizes the functional J ε,θ : We have the following result: Lemma 14. Under the assumptions of Theorem 2, the solutions m ε and f θ verify the following inequalities: ð Proof of Lemma 14. We denote by The functions yε and z θ verify Multiplying equality (108) and equality (109) by, respectively, y ε and z θ and integrating with respect to Q, we get Using the Young inequality, Cauchy-Schwarz inequality, and the fact that β is L ∞ , we prove that Therefore, choosing λ 0 > ðα 2 + + 3/2Þ, we get Finally, applying the result of Lemma 13 to the above inequality, it follows that and then the inequality (106) holds.

12
Abstract and Applied Analysis Likewise, we have Using the above inequality, Lemma 13, and inequality (115), we obtain and then we get the desired result.

Proof of Theorem 2.
In this section, we established the existence of a fixed point for the preceding auxiliary problem. Indeed, we consider that ðH 3 Þ holds and we suppose to simplify that λð0Þ = λðAÞ = 0: For each p ∈ L 2 ðQ T Þ, let us denote by ΛðpÞ ⊂ L 2 ð0, TÞ the set of all Ð A 0 λðaÞmðpÞda, where the couple ðmðpÞ, f ðpÞÞ is the solution of the following system: and ðnðpÞ, lðpÞÞ the corresponding solution of the minimizer of J ε,θ with mðpÞða, TÞ = f ðpÞða, TÞ = 0 for almost every a ∈ ð0, AÞ.
We have the following result.
Proposition 15. Under the assumptions of Theorem 2, for any p ∈ L 2 ðQ T Þ, the solution of problem (120) satisfies where YðtÞ = Ð A 0 λðaÞmðpÞda and the constant C is independent of p, m 0 , and f 0 : Proof of Proposition 15. Let YðtÞ = Ð A 0 λðaÞmðpÞda: It is easy to prove that Y is the solution of system where Using Lemma 14 and the assumptions on β and λ, we infer that there exists K > 0 such that By using (122), the Young inequality and integrating on Q T , we obtain Moreover, the Cauchy-Schwarz inequality leads to The inequality (105) and the fact that λ ∈ Cð½0, AÞ give where K 1 > 0 is independent of p, ε, and θ: Moreover, as λ μ m ∈ L 1 ð0, AÞ, and using (124), it follows that 13 Abstract and Applied Analysis Now, letỸ = e −λ 0 t Y: Then,Ỹ satisfies Multiplying the first equation of (129) byỸ, integrating on ð0, tÞ, and using successively Cauchy-Schwarz and Young inequalities, we deduce that Using the above calculations and choosing λ 0 > 2, we get The desired result comes from (128) and (131).
Applying the Kakutani fixed-point theorem [11] in the space L 2 ð0, TÞ to the mapping Λ, we infer that there is at least one Y ∈ Wð0, TÞ such that Y ∈ ΛðYÞ: This completes the null controllability proof of the model (1).

Proof of Theorem 3
4.3.1. Proof of . In this section, we always consider the following system: for every p in L 2 ðQ T Þ: Under the assumptions of Theorem 3, the controllability problem that is to find v m ∈ L 2 ðΘÞ such that ðm, f Þ solution of the system (133) verifies is equivalent to the following observability inequality.
For the proof of Proposition 16, we state the following estimate.
Proposition 17. Under the assumptions ðH 1 Þ and ðH 2 Þ, there exists a constant C > 0 such that the solution ðh, gÞ of system (136) verifies Moreover, we deduce for h T = 0 a.e in ð0, ϱÞ that there exists a constant C ϱ,T > 0 such that Proof of Proposition 16. Setting y = e λ 0 t g, the function y verifies Multiplying equality (139) by y and integrating on Q, we obtain Using Young inequality and the condition on β, we get Choosing δ = α 2 + , we obtain Finally, choosing λ 0 > α 2 + + 1, it follows that So, Finally, we get Combining the above inequality and inequality (44) of Proposition 8, for h T = 0 a.e in ð0, ρÞ, we get Proof of Proposition 17. We use the results of Proposition 9 and Proposition 17. Indeed, by combining (68) and (146), the desired result is obtained.
Now, let ε > 0 and ϱ > 0: We consider the functional J ε defined by 15 Abstract and Applied Analysis where ðm, f Þ is the solution of the following system: We have the following lemma.
Lemma 18. The functional J ε is continuous, strictly convex, and coercive. Consequently, J ε reaches its minimum at one and has v m,ε = χ Θ h ε and there exists positive constants C 1 , Proof of Lemma 18. The proof is similar to that of Lemma 13.
By making ε tending towards zero, we thus obtain that χ Θ h ε ⇀ χ Θ v m and ðmε, f ε Þ ⇀ ðm, f Þ, where ðm, f Þ is the solution of system (148) that verifies m :,T ð Þ= 0 a:e in ϱ, A ð Þ: Finally, a similar function Λ is defined and a similar procedure is followed to get the null controllability for the nonlinear problem. . Let p ∈ L 2 ðQ T Þ, under the assumptions of Theorem 3, the following controllability problem finds v f ∈ L 2 ðΘÞ such that the solution of the system is equivalent to the following observability inequality.

Proof of
Proposition 19. Let us assume true the assumptions ðH 1 Þ − ðH 2 Þ − ðH 3 Þ: For any T > a 1 + A − a 2 , there exists C T > 0 such that where g is a solution of the system Proof of Proposition 19. Using inequality (45) of Proposition 8, the result of Proposition 11, and the representation of the solution of the system (155), we get the desired result.
To conclude, a similar function Λ is defined and a similar procedure is followed to get the null controllability for the nonlinear system. We omit all details because the extension is straightforward.

Numerical Illustrations
In this part, the idea is to highlight the numerical simulation of the nonlinear problem.
The first parts are to reduce the PDE to the finite dimensional system of the form where A l , B l , and P l are matrices, and is the finite dimensional state vector. Here, vector P l is the contribution of the nonlinear part, which comes from births; B l is the control matrix, and Y l is the control vector.  We notice that numerically, we have the positivity of the state and the initial conditions, see Figures 3 and 4. Also, when we tend towards A, the density of each population tends towards zero. Therefore, this fact reflects the reality.
When the initial density of male population or female population is zero, we have no males or females during the evolution of the population, see Figures 1 and 2. This fact shows the importance of fertility function.
Moreover, in the absence of males or females, we obtain the total extinction of females or males at time t = A = 2 confirming the reality.
We use ODE 45 for all the simulations of the system.

Example 2.
We construct the control problem, which consists in minimizing the function, and we choose the classical Hum function The approximate null controllability becomes the minimization of the functional J ε , where ðU, YÞ verifies the system dU l dt = A l U l + P l + B l Y l : In this example, we use the data of Example 1. For the final time T = 0:5, the control states and the controlled states are given below.
We notice that the uncontrolled solutions are not null at the corresponding final time (see Figure 4) while the controlled states are zero at the corresponding final time (see Figure 5). However, we could not take a positivity constraint in our simulations (see Figure 6).
The CaSadi toolbox is used to simulate the control system (by the minimization of the function: Hum method).

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.