# Null Controllability of a Four Stage and Age-Structured Population Dynamics Model

Authors: Amidou Traoré; Bedr’Eddine Ainseba; Oumar Traoré
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper is devoted to study the null controllability properties of a population dynamics model with age structuring and nonlocal boundary conditions. More precisely, we consider a four-stage model with a second derivative with respect to the age variable. The null controllability is related to the extinction of eggs, larvae, and female population. Thus, we estimate a timeT to bring eggs, larvae, and female subpopulation density to zero. Our method combines fixed point theorem and Carleman estimate. We end this work with numerical illustrations.


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## Body

## 1. Introduction

Letuit,a,1≤i≤4, be, respectively, the distribution of eggs, larvae, and female and male individuals of age a at time t; Ai,1≤i≤4, is the life expectancy of an i−stage individual and T is a positive constant. We consider the following population dynamics model based on Fokker–Planck or Kolmogorov-type equations which is written as(1)∂tu1t,a+∂av1ζt,au1t,a−k∂a2u1t,a=−μ1ζt,a+β1ζt,au1t,a+maw1t,a,∂tu2t,a+∂av2ζt,au2t,a−k∂a2u2t,a=−μ2ζt,a+β2ζt,au2t,a+maw2t,a,∂tu3t,a+∂av3ζt,au3t,a−k∂a2u3t,a=−μ3ζt,au3t,a+maw3t,a,∂tu4t,a+∂av4ζt,au4t,a−k∂a2u4t,a=−μ4ζt,au4t,a,where t,a∈Ωi=0,T×0,Ai and w1,w2, and w3 are, respectively, the control function of eggs, larvae, and females. Here, ma is the characteristic function of ω=0,a∗ with 0<a∗<minA1,A2,A3. To complete the system, the boundary conditions are stated as follows:(2)v1ζt,au1t,a−k∂au1t,aa=0=∫0A3β3ζt,au3t,ada,v2ζt,au2t,a−k∂au2t,aa=0=∫0A1β1ζt,au1t,ada,v3ζt,au3t,a−k∂au3t,aa=0=∫0A2σβ2ζt,au2t,ada,v4ζt,au4t,a−k∂au4t,aa=0=∫0A21−σβ2ζt,au2t,ada,ui0,a=ui0a,a∈0,Ai,1≤i≤4,uit,Ai=0,t∈0,T,1≤i≤4.Here,μiζt,a,βiζt,a, and viζt,a are, respectively, the i−stage mortality rate, the i−stage age specific transition functions, and the i−stage growth rate of age a at temperature ζt and at time t.Finally,(3)∫0Aiσiβiζt,auit,ada,1≤i≤4,σ1=1,σ2=σ,σ3=1,σ4=1−σ.Denote, respectively, the distribution of the newborns of eggs, larvae, females, and males. The constantσ denotes the sex ratio and k>0 (diffusion coefficient) corresponds to small deviations of order 2 which represent the dispersion effects of individuals during their development. Our aim is to study the null controllability of (1) and (2).Given a fixedT>0, we look for controls w1,w2, and w3 such that u1,u2, and u3 satisfy(4)u1T,a=u2T,a=u3T,a=0.In practice, this study applies to the dynamics cowpea pest insects (Callosobruchus maculatus). These insects live in hot areas, namely, tropical and subtropical regions [1] and cause important damage to the seeds of cowpeas and other legumes. This damage is exclusively due to the larvae. Indeed, females lay their eggs on the pods or on stocked seeds and then larvae grow by feeding on the cotyledons, see [2]. Twenty five days and a temperature varying between twenty-seven and thirty-one degrees Celsius are necessary for the growth of the larvae.The life cycle ofCallosobruchus maculatus can be divided into three development stages: egg, larva + pupa (growing stages), and adult (reproduction ability). Besides, during their growth, beetle larvae transform nitrogen to toxic uric acid that accumulates into cowpea and make it unfit for consumption.Several methods of combating these insects have been considered. Among them is the method of preservation in hermetically sealed containers, thus preventing the development of the larvae.The method of control we are considering is to remove or eliminate individuals. This can be done with the help of pesticides. We recall that the optimal and exact control problems are widely investigated for age-structured population dynamics by many researchers. Most of these studies are focused on optimal control problems [3–5] and the references therein. One can also refer to [6, 7]. Lebeau and Robbiano establish in [8] the null controllability of the linear heat equation. Next, the exact controllability for age dependent linear and nonlinear single-species population models with spatial diffusion was investigated by Aïnsebaet al. [9, 10]. Barbu et al. also considered the exact controllability of the linear Lotka–McKendrick model without spatial structure by establishing an observability inequality for the backward adjoint system [11]. Later on, He and Ainseba investigate the exact null controllability of a stage and age-structured population dynamics system in [12] and the exact null controllability of the Lobesia Botrana model with spatial diffusion in [13].Moreover, in [14], the authors investigated a semilinear problem: the null controllability of the heat equation with a Fourier boundary condition over the entire boundary. In our work, the Fourier condition contains a nonlocal term and is set only on a part of the domain. This induces additional difficulties in establishing the Carleman inequality.In [15], Hegoburu and Anita study in a very practical way the null controllability of a nonlinear model of population dynamics by means of a feedback control. This interesting result cannot be applied easily in our case since we study a four-stage model via three controls. More precisely, in our case, we control only three stages. So, it seems difficult here to compute explicitly the feedback control as it was done in [15].The fundamental difference with [13] lies in the fact that our model combines a transport effect with the presence of a first-order derivation with respect to the age variable and a diffusion along the same variable. Thus, the inequalities guaranteeing the null controllability obtained in [13] become inoperative in our case.As far as we know, there are no results dealing with the null controllability for an age and stage-dependent system involving a second derivative with respect to the age variable. The second derivative with respect to the physiological age variable corresponds to small deviations of order two and represents the dispersion effects of individuals during their development due to, for example, the temperature, the humidity, and the quality of food. This article is structured as follows. In Section2, the assumptions and the main result are stated. We study the null controllability of some associated auxiliary model in Section 3. Section 4 is devoted to the proof of the main result, and some numerical illustrations are given in Section 5.


## 2. Assumptions and Main Result

For the sequel, we make the following assumptions on the demographic parameters:(A1) The transition functionsβkζ.,. are positive and βk∈L∞0,T×0,Ak,1≤k≤3.(A2) The mortality functionsμkζ.,. are positive and μk∈L∞0,T×0,Ak,1≤k≤4.(A3) The initial conditionsuk0a≥0 a.e. a∈0,A,1≤k≤4.(A4) The growth functionsviζ.,.,1≤i≤4 are positive and bounded, that is,(5)0<vimin<viζt,a<vimax,∀t,a∈0,T×0,Ai.In addition, for allt∈0,T, viζt,.∈C10,A, and there is a positive constant Ci such that(6)∂aviζt,a∞≤Ci,∀t,a∈0,T×0,Ai.(A5) The temperature functionζ. is measurable.These assumptions are biologically meaningful, see [3, 16–18].Settingu^i=e−λtui,1≤i≤4, the vector u^1,u^2,u^3,u^4 verifies the system:(7)∂tu^1+∂av1u^1−k∂a2u^1=−μ1+β1+λu^1+maw^1,∂tu^2+∂av2u^2−k∂a2u^2=−μ2+β2+λu^2+maw^2,∂tu^3+∂av3u^3−k∂a2u^3=−μ3+λu^3+maw^3,∂tu^4+∂av4u^4−k∂a2u^4=−μ4+λu^4,v1ζt,au^1t,a−k∂au^1t,aa=0=∫0A3β3ζt,au^3t,ada,v2ζt,au^2t,a−k∂au^2t,aa=0=∫0A1β1ζt,au^1t,ada,v3ζt,au^3t,a−k∂au^3t,aa=0=∫0A2σβ2ζt,au^2t,ada,v4ζt,au^4t,a−k∂au^4t,aa=0=∫0A21−σβ2ζt,au^2t,ada,u^i0,a=u^0a,a∈0,Aiandu^it,Ai=0,t∈0,T,1≤i≤4.Now, we consider the following auxiliary system:(8)∂tu^1+∂av1u^1−k∂a2u^1=−μ1+β1+λu^1+maw^1,∂tu^2+∂av2u^2−k∂a2u^2=−μ2+β2+λu^2+maw^2,∂tu^3+∂av3u^3−k∂a2u^3=−μ3+λu^3+maw^3,∂tu^4+∂av4u^4−k∂a2u^4=−μ4+λu^4,(9)v1ζt,au^1t,a−k∂au^1t,aa=0=∫0A3β3ζt,aϕ3t,ada,v2ζt,au^2t,a−k∂au^2t,aa=0=∫0A1β1ζt,aϕ1t,ada,v3ζt,au^3t,a−k∂au^3t,aa=0=∫0A2σβ2ζt,aϕ2t,ada,v4ζt,au^4t,a−k∂au^4t,aa=0=∫0A21−σβ2ζt,aϕ2t,ada,u^i0,a=u^0a,a∈0,Aiandu^it,Ai=0,t∈0,T,1≤i≤4.For the existence of the solutions to problems similar to equations (8) and (9), we refer, for instance, to [19] and the references therein.LetA=maxA1,A2,A3,A4, we can state the main result.Theorem 1.
Assume thatA1−A5 hold. Let A>0 and T>0 be given. Then, for all u10,u20,u30∈L20,A3, there exist a control w=w1,w2,w3∈L20,T×ω3 such that the associated solution of equations (1) and (2) verify (4).


## 3. Null Controllability of an Auxiliary System

This section is devoted to the null controllability of the following auxiliary system obtained from equations (1) and (2):(10)∂tu1t,a+∂av1ζt,au1t,a−k∂a2u1t,a=−μ1ζt,a+β1ζt,au1t,a+maw1t,a,∂tu2t,a+∂av2ζt,au2t,a−k∂a2u2t,a=−μ2ζt,a+β2ζt,au2t,a+maw2t,a,∂tu3t,a+∂av3ζt,au3t,a−k∂a2u3t,a=−μ3ζt,au3t,a+maw3t,a,∂tu4t,a+∂av4ζt,au4t,a−k∂a2u4t,a=−μ4ζt,au4t,a,(11)v1ζt,au1t,a−k∂au1t,aa=0=h1t,v2ζt,au2t,a−k∂au2t,aa=0=h2t,v3ζt,au3t,a−k∂au3t,aa=0=h3t,v4ζt,au4t,a−k∂au4t,aa=0=h4t,ui0,a=ui0a,a∈0,A,1≤i≤4,uit,a=0,t∈0,T,1≤i≤4.Here,hi=∫0Aσiβjζt,aϕjt,ada,i,σi,j∈1,1,3,2,1,1,3,σ,2,4,1−σ,2. We first established a Carleman-type inequality for the following adjoint system of equations (10) and (11):(12)−∂tq1−∂av1q1−k∂a2q1+μ1+β1+∂av1q1=0,−∂tq2−∂av2q2−k∂a2q2+μ2+β2+∂av2q2=0,−∂tq3−∂av3q3−k∂a2q3+μ3+∂av3q3=0,−∂tq4−∂av4q4−k∂a2q4+μ4+∂av4q4=0,with t,a∈Q and the following boundary conditions(13)qit,A=∂aqit,0=0,t∈0,T,1≤i≤4;qiT,a=−uiT,aε,a∈0,A,1≤i≤3;q4T,a=0,a∈0,A.The system equations (12) and (13) admit a unique solution.Letω0⊂⊂ω be a nonempty bounded subset of Ω=0,A, and ψ∈C2Ω¯ satisfying (see Lemma 1.1 in [20])(14)ψa>0;for anya∈Ω,ψa=0,for anya∈∂Ω,∇ψa>0,for anya∈Ω∖ω0¯.Now, we define the following weight functions, which are constructed using the functionψ (see [21] or [20]):(15)φt,a=eλψatT−t,αt,a=−e2λψCΩ¯+eλψatT−t,φ˜t,a=e−λψatT−t,α˜t,a=−e2λψCΩ¯+e−λψatT−t,where λ is an appropriate positive constant.Remark 1.
Using the definition ofφ,φ˜,α, and α˜, we deduce that there is a positive constant c such that(1)
∂aφ=λ∂aψφ,∂aφ˜=−λ∂aψφ˜,∂aα=λ∂aψφ,and∂aα˜=−λ∂aψφ˜(2)
∂tφ≤cφ2,∂tφ˜≤cφ˜2,∂tα≤cφ2,∂tα˜≤cφ˜2,∂t2φ≤cφ3,∂t2φ˜≤cφ˜3,∂t2α≤cφ3,∂t2α˜≤cφ˜3,∂t∂aα≤cφ2,and∂t∂aα˜≤cφ˜2Lemma 1.
Letφ,α,φ˜, and α˜ be defined as in equation (15). Then, there exists a number λ^>1 such that, for any λ>λ^, there exists a number s0λ such that, for every s≥s0λ, we have the following inequalities:(16)∫Q1sφ∂tqi2+∂a2qi2+sφ∂aqi2+s3φ3qi2e2sα+e2sα˜dtdx≤C∫0,T×ωs3φ3qi2e2sα+e2sα˜dtda,for all qi∈L20,T×0,A solution of (12) and (13), i=1,2,3, and(17)∫Qφ3q12+q22+q32e2sα+e2sα˜dtda≤C∫0,T×ωφ3q12+q22+q32e2sα+e2sα˜dtda.Remark 2.
Here, we have to control the boundary conditions which is not the case studied in [20].Proof.
of Lemma1. Let us consider the operator(18)L^qi=−∂tqi−k∂a2qi,1≤i≤3.
We set(19)g˜it,a=−μi+βiqi+vi∂aqi,β3=β4=0,1≤i≤3.
Letwi=esαqi and w˜i=esα˜qi,1≤i≤3, where s is a positive constant. Put Σ0=0,T×A, Σ1=0,T×0, and Σ=Σ0∪Σ1.
From (15), we have(20)wiT,a=w˜iT,a=wi0,a=w˜i0,a=0on0,A.
Let us define the operatorsP and P˜ as follows: Pwi=esαL^e−sαwiandP˜wi=esα˜L^e−sα˜wi.
It follows from equations (12), (18), and (19) that(21)Pwi=esαL^e−sαwi=esαg˜i,onQ,P˜w˜i=esα˜L^e−sα˜w˜i=esα˜g˜i,onQ.
Explicitly, we have(22)Pwi=−∂twi−k∂a2wi+s∂tαwi+2ksλ∂aψφ∂awi+ksλ∂a2ψφwi+ksλ2∂aψ2φwi−ks2λ2∂aψ2φ2wi,(23)P˜w˜i=−∂tw˜i−k∂a2w˜i+s∂tα˜w˜i−2ksλ∂aψφ˜∂aw˜i−ksλ∂a2ψφ˜w˜i+ksλ2∂aψ2φ˜w˜i−ks2λ2∂aψ2φ2w˜i.
We also consider the operatorsL1,L2,L˜1, and L˜2 defined as follows:(24)L1wi=−k∂a2wi−ks2λ2∂aψ2φ2wi+s∂tαwi,L2wi=−∂twi+2ksλ∂aψφ∂awi+2ksλ2∂aψ2φwi,L˜1w˜i=−k∂a2w˜i−ks2λ2∂aψ2φ˜2w˜i+s∂tα˜w˜i,L˜2w˜i=−∂tw˜i−2ksλ∂aψφ˜∂aw˜i+2ksλ2∂aψ2φ˜w˜i.
Using equations (19), (22), and (24), we obtain(25)L1wi+L2wi=fs,inQ,L˜1w˜i+L˜2w˜i=f˜s,inQ,where(26)fs=g˜iesα−ksλφ∂a2ψwi+ksλ2φ∂aψ2wi,f˜s=g˜iesα˜+ksλφ˜∂a2ψw˜i+ksλ2φ˜∂aψ2w˜i.
Taking theL2-norm of (25), we obtain(27)fsL2Q2=L1wiL2Q2+L2wiL2Q2+2L1wi,L2wiL2Q,f˜sL2Q2=L˜1w˜iL2Q2+L˜2w˜iL2Q2+2L˜1w˜i,L˜2w˜iL2Q.
By using (24), we have the following equalities:(28)2L1wi,L2wiL2Q=2k2s3λ4∫Qwi2φ3∂aψ4dtda+6k2sλ2∫Q∂awi2φ∂aψ2dtda−2ksλ∫0Twi∂twiφ∂aψa=0dt+4k2s2λ3∫0Twi2φ2∂aψ3a=0dt+2k2s3λ3∫0Twi2φ3∂aψ3a=0dt−2k2sλ∫0T∂awi2φ∂aψ0Adt−2ks2λ∫0Twi2φ∂aψ∂tαa=0dt+X1,(29)2L˜1w˜i,L˜2w˜iL2Q=2k2s3λ4∫Qw˜i2φ˜3∂aψ4dtda+6k2sλ2∫Q∂aw˜i2φ˜∂aψ2dtda+2ksλ∫0Tw˜i∂tw˜iφ˜∂aψa=0dt−4k2s2λ3∫0Tw˜i2φ˜2∂aψ3a=0dt−2k2s3λ3∫0Tw˜i2φ˜3∂aψ3a=0dt+2k2sλ∫0T∂aw˜i2φ˜∂aψ0Adt+2ks2λ∫0Tw˜i2φ˜∂aψ∂tα˜a=0dt+X2,with(30)X1≤c1∫Qs3λ3φ3+s2λ4φ3wi2+sλφ+1∂awi2dtda,∀s≥1,λ≥1,(31)X2≤c2∫Qs3λ3φ˜3+s2λ4φ˜3w˜i2+sλφ˜+1∂aw˜i2dtda,∀s≥1,λ≥1.
Considering equations (27)–(29) and using the fact that ψa=0 on ∂Ω, it follows that(32)fsL2Q2+f˜sL2Q2=L1wiL2Q2+L2wiL2Q2+L˜1w˜iL2Q2+L˜2w˜iL2Q2+2X1+2X2+2k2s3λ4∫Qwi2φ3∂aψ4+w˜i2φ˜3∂aψ4dtda+6k2sλ2∫Q∂awi2φ∂aψ2+∂aw˜i2φ˜∂aψ2dtda,and we deduce from (32)(33)fsL2Q2+f˜sL2Q2≥L1wiL2Q2+L2wiL2Q2+L˜1w˜iL2Q2+L˜2w˜iL2Q2+X1+X2+2k2s3λ4∫Qwi2φ3∂aψ4+w˜i2φ˜3∂aψ4dtda+2k2sλ2∫Q∂awi2φ∂aψ2+∂aw˜i2φ˜∂aψ2dtda.
Note that there existsδ>0 such that(34)∂aψa>δ>0,∀a∈Ω∖ω0¯.
After these calculations, we can now use the same arguments developed in [21] or [20] to get (16).
By summing with respect toi, one deduces easily inequality (17). We can now establish the observability inequality of the system equations (12) and (13) and then the null controllability result of the system equations (10) and (11). We start by proving our observability inequality by means of Carleman estimate.Lemma 2.
We have the following inequality:(35)∫0A∑i=13qi0,a2da+∫0T∑i=13qi0,a2dt≤K∫0,T×ω∑i=13qi2dtda,for all solution, q=q1,q2,q3∈L20,T×0,A3 of equations (12) and (13).Letλ be a positive constant and q^it,a=eλtqit,a,1≤i≤4. Then, equations (12) and (13) become(36)−∂tq^1−∂av1q^1−k∂a2q^1+λ+μ1+β1+∂av1q^1=0,−∂tq^2−∂av2q^2−k∂a2q^2+λ+μ2+β2+∂av2q^2=0,−∂tq^3−∂av3q^3−k∂a2q^3+λ+μ3+∂av3q^3=0,−∂tq^4−∂av4q^4−k∂a2q^4+λ+μ4+∂av4q^4=0,(37)q^it,A=∂aq^it,0=0,t∈0,T,1≤i≤4;q^iT,a=−eλTuiT,aε,a∈0,A1≤i≤3;q^4T,a=0,a∈0,A.For the proof of Lemma2, we need an auxiliary result. This result is given in the following proposition.Proposition 1.
Letq^=q^1,q^2,q^2∈L20,T×0,A3 be solution of (36) and (37). Then,(38)∫0Aq^10,a2+q^20,a2+q^30,a2da≤K′∫0,T×ωq^12+q^22+q^32dtda.Proof.
of Proposition1. Remark that(39)φ3e2sα˜≥e−2C1+1/T2T6,t∈T/4,3T/4,(40)φ3e2sα≤B1e3C,t∈0,T.
Using (39), (40), and (16), it follows that(41)∫Ω×T/4,3T/4q^i2dtda≤K1∫0,T×ωq^i2dtda.
Letθ∈C20,T be a cut-off function such that(42)θt=1t∈0,T/4,0≤θt≤1t∈T/4,3T/4,θt=0t>3T/4.
Multiplying the equationi of (36) by θq^i and integrating by parts on Q, we obtain(43)∫0Aq^i0,a2da+2λ−Ci∫Qq^i2θdtda≤∫Q−∂tθq^i2dtda.
From the definition ofθ, it follows that(44)∫0Aq^i0,a2da≤K2∫Ω×T/4,3T/4q^i2dtda.
Next, (41) and (44) give(45)∫0Aq^i0,a2da≤K′∫0,T×ωq^i2dtda.
Finally, (45) leads to the desired result by summing with respect to i,1≤i≤3.Proof.
of Lemma2.
Note thatqit,0=−∫0A∂aqida. Using Cauchy–Schwarz and Young inequalities and integrating on 0,T, we obtain(46)∫0Tqit,02dt≤AρT/2∫0T∫0Aρ∂aqi2dadt.
By using (16), (40), and (46), it follows that(47)∫0Tqit,02dt≤Ks∫0,T×ωqi2dtda,with Ks=s3B1e3C. Now, as q^i0,a=qi0,a,q^i2≤e2λTqi2, using (47) and Proposition 1, we get (35).
With our observability inequality, we are ready to prove the null controllability result.Theorem 2.
Assume thatA1−A5 hold. Let A>0 and T>0 be given.Then, for allu10,u20,u30∈L20,A3, there exists a control w=w1,w2,w3∈L20,T×ω3 such that the associated solution of (10) and (11) verifies (4).Proof.
of Theorem2.
We consider systems (10) and (11) and the optimal control problem:(48)Minimize12∫Qw1t,a2+w2t,a2+w3t,a2dtda+12ε∫0Au1T,a2+u2T,a2+u3T,a2da,where Q=0,T×0,A,wi∈L2Q, and ui1≤i≤3 are solutions of (10) and (11).
Let(49)Jεw=12∫Qw1t,a2+w2t,a2+w3t,a2dtda+12ε∫0Au1T,a2+u2T,a2+u3T,a2da,with w=w1,w2,w3.
The functionalJε is continuous, convex, and coercive. Hence, it admits a unique minimizer wε=w1,ε,w2,ε,w3,ε, and we have after easy computations(50)wi,εt,a=maqi,εt,a,a.e.t,a∈Q,1≤i≤3.
Note thatq1,ε,q2,ε,q3,ε is the solution of (12) and (13) and uε=u1,ε,u2,ε,u3,ε is the solution of (10) and (11) associated to wε.
Multiplying equation numberi of (12) and (13) by ui,ε and integrating on Q, it follows that(51)∫Qqi,εmawi,εdtda+1ε∫0Aui,ε2T,ada+∫0Aqi,ε0,aui0ada+∫0Tqi,εt,0hitdt=0.
From (51) and (50), we obtain(52)∫0,T×ωwi,ε2dtda+1ε∫0Aui,ε2T,ada=−∫0Aqi,ε0,aui0ada−∫0Tqi,εt,0hitdt.
Young inequality in (52) leads to(53)∫0,T×ωwi,ε2dtda+1ε∫0Aui,ε2T,ada≤12K∫0Aqi,ε20,ada+∫0Tqi,ε2t,0dt+2K∫0Aui02ada+∫0Thi2tdt.
Thus,(54)∫0,T×ω∑i=13wi,ε2dtda+1ε∫0A∑i=13ui,ε2T,ada≤12K∫0A∑i=13qi,ε20,ada+∫0T∑i=13qi,ε2t,0dt+2K∫0A∑i=13ui02ada+∫0T∑i=13hi2tdt.
Therefore, inequalities (35) and (54) imply(55)12∫0,T×ω∑i=13wi,ε2dtda+1ε∫0A∑i=13ui,ε2T,ada≤2K∫0A∑i=13ui02ada+∫0T∑i=13hi2tdt.
From (55), we deduce(56)wi,εL20,T×ω2≤4K∫0A∑i=13ui02ada+∫0T∑i=13hi2tdt,1≤i≤3,∫0Aui,ε2T,ada≤2εK∫0A∑i=13ui02ada+∫0T∑i=13hi2tdt,1≤i≤3.
Then, one can extract subsequences also denoted bywi,ε,ui,ε1≤i≤3 such that wi,ε⟶wi weakly in L20,T×ω and ui,ε⟶ui weakly in L20,T×0,A. Moreover, u=u1,u2,u3 is the unique solution of (10) and (11) associated to w=w1,w2,w3 and verifies (4).


## 4. Proof of the Main Result

Now, we consider a real numberλ0>0 and we set u˜i=e−λ0tui1≤i≤4. Then, u˜=u˜1,u˜2,u˜3,u˜4 is solution of the following system:(57)∂tu˜1+∂av1u˜1−k∂a2u˜1+λ0+μ1+β1u˜1=maw˜1,∂tu˜2+∂av2u˜2−k∂a2u˜2+λ0+μ2+β2u˜2=maw˜2,∂tu˜3+∂av3u˜3−k∂a2u˜3+λ0+μ3u˜3=maw˜3,∂tu˜4+∂av4u˜4−k∂a2u˜4+λ0+μ4u˜4=0,with t,a∈Ω=0,T×0,A,w˜i=e−λ0twi, and(58)v1u˜1−k∂au˜1a=0=∫0Aβ3u˜3t,ada,v2u˜2−k∂au˜2a=0=∫0Aβ1u˜1t,ada,v3u˜3−k∂au˜3a=0=∫0Aβ2u˜2t,ada,v4u˜4−k∂au˜4a=0=∫0A1−σβ2u˜2t,ada,u˜i0,a=u˜i0a,a∈ΩA=0,A,1≤i≤4,u˜it,A=0,t∈ΩT=0,T,1≤i≤4.Let us consider the following system:(59)∂tu˜1+∂av1u˜1−k∂a2u˜1+λ0+μ1+β1u˜1=maw˜1,∂tu˜2+∂av2u˜2−k∂a2u˜2+λ0+μ2+β2u˜2=maw˜2,∂tu˜3+∂av3u˜3−k∂a2u˜3+λ0+μ3u˜3=maw˜3,∂tu˜4+∂av4u˜4−k∂a2u˜4+λ0+μ4u˜4=0,and(60)v1u˜1−k∂au˜1a=0=h1t,v2u˜2−k∂au˜2a=0=h2t,v3u˜3−k∂au˜3a=0=h3t,v4u˜4−k∂au˜4a=0=h4t,u˜i0,a=u˜i0a,a∈ΩA=0,A,1≤i≤4,u˜it,A=0,t∈ΩT=0,T,1≤i≤4,with h1,h2,h3,h4∈L2ΩT is defined as in (58).Note that, from Section3, systems (59) and (60) are null controllable. Now, we will prove the null controllability of (57) and (58).Letℳ=∏i=14L2ΩT, and we define the following set:(61)AγR1,R2,R3,R4=w˜=w˜1,w˜2,w˜3∈∏i=13L2q,u˜is solution of59−60andu˜1,u˜2,u˜3verifies4;hi=Riandw˜satisfies56.We introduce a multivalued function as follows:(62)Λγ:R1,R2,R3,R4∈ℳ↦ΛγR1,R2,R3,R4∈2ℳ,such that(63)ΛγR1,R2,R3,R4=∫0Aβ3u˜3,w˜t,ada,∫0Aβ1u˜1,w˜t,ada,∫0Aσβ2u˜2,w˜t,ada,∫0A1−σβ2u˜2,w˜t,ada:u˜1,w˜,u˜2,w˜,u˜3,w˜is solution of59−60andw˜∈AγR1,R2,R3,R4.The goal is to prove that the multivalued functionΛγ admits a fixed point. To this end, we will use the generalization of Leray–Schauder fixed point theorem. Consider(64)Nγ=R1,R2,R3,R4∈ℳ:∃ρ∈0,1,R1,R2,R3,R4∈ρΛγR1,R2,R3,R4.The fixed point existence of the multivalued functionΛγ is an immediate consequence of the following proposition.Proposition 2.
(i)
Nγ is bounded on ℳ(ii)
For allR1,R2,R3,R4∈ℳ,ΛγR1,R2,R3,R4 is a nonempty closed and convex subset of ℳ(iii)
Λγ:ℳ⟶2ℳ is compact(iv)
The multivalued mappingΛγ is upper semicontinuous on ℳProof.
of Proposition2.
Let us setu˜1,w˜,u˜2,w˜,u˜3,w˜,u˜4,w˜=u˜1,u˜2,u˜3,u˜4. i Let R1,R2,R3,R4∈Nγ. There is ρ∈0,1 such that 1/ρR1,R2,R3,R4∈ΛγR1,R2,R3,R4. Hence, there is u˜1,u˜2,u˜3∈∏i=13L2Ω associated to w˜∈∏i=13L2q such that(65)R1=∫0Aβ3u˜3da,R2=∫0Aβ1u˜1da,R3=∫0Aβ2u˜2da,R4=∫0Aβ2u˜2da,with u˜1,u˜2,u˜3 solution of (57) and (58) associated to w˜ satisfying (56). Then, we have(66)∑i=14RiL2ΩT2≤∑i=14ξiu˜iL2Ω2ξi=AβiL∞0,Aandξ4=0.
Multiplying the equation numberi of (57) and (58) by u˜i and integrating by parts on Ω, we obtain after summing with respect to i1≤i≤4:(67)∑i=142λ0−Ci−1u˜iL2Ω2≤∑i=14u˜i0L20,A2+∑i=14RiL2ΩT2+∑i=14w˜iL2q2,w4≡0.
In virtue of (66), (67), and (56), it follows that(68)∑i=142λ0−Ci−1u˜iL2Ω2≤4K+1∑i=14u˜i0L20,A2+4K+1∑i=14ξiu˜iL2Ω2.
Inequality (68) leads to(69)∑i=142λ0−Ci−1−4K+1ξiu˜iL2Ω2≤4K+1∑i=14u˜i0L20,A2.
Considering again (66) and (69), we get the desired result. iiR1,R2,R3,R4∈ℳ,ΛγR1,R2,R3,R4 is nonempty. In fact, systems (57) and (58) are equivalent to systems (10) and (11) which admits a solution. Also, w˜ satisfies (56).
Since the functionR1,R2,R3,R4↦u˜1,u˜2,u˜3,u˜4 is affine, it follows that ΛγR1,R2,R3,R4 is convex.
Letηn1,ηn2,ηn3,ηn4∈ΛγR1,R2,R3,R4 such that ηn1,ηn2,ηn3,ηn4⟶η1,η2,η3,η4 in ℳ. Thus, for all n, there exists w˜n which satisfies (56), such that(70)ηn1=∫0Aβ3u˜3,nda,ηn2=∫0Aβ1u˜1,nda,ηn3=∫0Aσβ2u˜2,nda,ηn4=∫0A1−σβ2u˜2,nda.
Here,u˜1,n,u˜2,n,u˜3,n is solution of (59) and (60), with R1,R2,R3,R4 instead of h1,h2,h3,h4 and u˜1,n,u˜2,n,u˜3,n which verify (69). From (66), (69), and (56), one can extract sequences still denoted by u˜1,n,u˜2,n,u˜3,nn and w˜nn that converge weakly towards u˜1,u˜2,u˜3 and w˜ in L2Ω3 and in L2q3, respectively.
So,(71)η1=∫0Aβ3u˜3da,η2=∫0Aβ1u˜1da,η3=∫0Aσβ2u˜2da,η4=∫0A1−σβ2u˜2da.
Moreover,u˜1,u˜2,u˜3 is solution of (59) and (60), with R1,R2,R3,R4=h1,h2,h3,h4. Also, w˜ verifies (56) and u˜1,u˜2,u˜3 verifies (4). Then, η1,η2,η3,η4∈ΛγR1,R2,R3,R4. iii Let R1,R2,R3,R4∈K a bounded subset of ℳ. Then, there exist ri>01≤i≤4 such that(72)RiL2QT≤ri,1≤i≤4.
Considerηn1,ηn2,ηn3,ηn4n∈ΛγR1,R2,R3,R4; for all n∈ℕ, there is u˜1,n,u˜2,n,u˜3,n∈L2Ω3 and a control function w˜n∈L2q3 such that(73)ηn1=∫0Aβ3u˜3,nda,ηn2=∫0Aβ1u˜1,nda,ηn3=∫0Aσβ2u˜2,nda,ηn4=∫0A1−σβ2u˜2,nda,where u˜1,n,u˜2,n,u˜3,n is solution of (59) and (60) with R1,n,R2,n,R3,n,R4,n instead of h1,h2,h3,h4 and w˜n verifies (56). Using (56), it follows that(74)∫q∑i=13w˜i,n2t,adtda≤4K∫0A∑i=14u˜i02da+∑i=14ri2.
Therefore,w˜n is bounded, and we can extract a sequence still denoted by w˜n that weakly converges towards w˜ in L2q.
Now, multiplying the equation numberi of (59) and (60) by u˜i,n, integrating by parts on Q and using (72), we get after summing(75)∫Q∑i=14ku˜i,na2t,a+u˜i,n2t,adtda≤4K+1∫0A∑i=14u˜i02da+∑i=14ri2.
Consequently, the sequenceu˜1,n,u˜2,n,u˜3,n is bounded in L2Q3. One can extract a sequence denoted u˜1,nk,u˜2,nk,u˜3,nk that converges weakly towards u˜1,u˜2,u˜3 in L2Q3.
As(76)ηnk1=∫0Aβ3u˜3,nkda,ηnk2=∫0Aβ1u˜1,nkda,ηnk3=∫0Aσβ2u˜2,nkda,ηnk4=∫0A1−σβ2u˜2,nkda,using (75), we obtain(77)∑i=14ηnkiL2QT≤K′∫0A∑i=14u˜i02da+∑i=14ri2.
From (77), one can extract a sequence which also denoted ηnk1,ηnk2,ηnk3,ηnk4 that converges weakly to η1,η2,η3,η4 on L2QT.
So,(78)∫QTϕηnjidt⟶∫QTϕηidt,∀ϕ∈L2QT,1≤i≤4.
Moreover, the sequenceu˜1,nk,u˜2,nk,u˜3,nk converges weakly towards u˜1,u˜2,u˜3 on L2Q3. Thus, the sequence u˜1,nk,u˜2,nk,u˜3,nk associated to ηnk1,ηnk2,ηnk3,ηnk4 converges weakly towards u˜1,u˜2,u˜3. Since ϕβk∈L2QT, for all ϕ∈L2QT and for all k∈1,2,3, it follows that(79)∫QTϕηnk1dt⟶∫QTϕ∫0Aβ3u˜3dadt,∫QTϕηnk2dt⟶∫QTϕ∫0Aβ1u˜1dadt,∫QTϕηnk3dt⟶∫QTϕ∫0Aσβ2u˜2dadt,∫QTϕηnk4dt⟶∫QTϕ∫0A1−σβ2u˜2dadt.
Equations (78) and (79) imply(80)∫QTϕη1−∫0Aβ3u˜3dadt=0,∀ϕ∈L2QT,∫QTϕη2−∫0Aβ1u˜1dadt,∀ϕ∈L2QT,∫QTϕη3−∫0Aσβ2u˜2dadt=0,∀ϕ∈L2QT,∫QTϕη4−∫0A1−σβ2u˜2dadt=0,∀ϕ∈L2QT.
Consequently,(81)η1=∫0Aβ3u˜3da,a.e.t∈QT,η2=∫0Aβ1u˜1da,a.e.t∈QT,η3=∫0Aσβ2u˜2da,a.e.t∈QT,η4=∫0A1−σβ2u˜2da,a.e.t∈QT.
Then,u˜1,u˜2,u˜3,u˜4 is solution of (59) and (60) with R1,R2,R3,R4 instead of h1,h2,h3,h4; u˜1,u˜2,u˜3 and w˜ satisfy, respectively, (4) and (56). iv Let K be a closed subset of ℳ. We will prove that Λγ−1K is a closed subset of ℳ. Remark that Λγ−1K=R1,R2,R3,R4∈ℳ;ΛγR1,R2,R3,R4∩K≠∅.
Suppose thatR1,n,R2,n,R3,n,R4,n⟶R1,R2,R3,R4 (strongly) in ℳ. Then, the sequence R1,n,R2,n,R3,n,R4,n is bounded. There also exists a sequence ηn1,ηn2,ηn3,ηn4n∈K, such that, for all n≥1,ηn1,ηn2,ηn3,ηn4∈ΛγR1,n,R2,n,R3,n,R4,n. So, there exist u˜1,n,u˜2,n,u˜3,n∈L2Ω3 and a control function w˜n∈L2q3 such that(82)ηn1=∫0Aβ3u˜3,nda,ηn2=∫0Aβ1u˜1,nda,ηn3=∫0Aσβ2u˜2,nda,ηn4=∫0A1−σβ2u˜2,nda,with w˜n, u˜1,n,u˜2,n,u˜3,n verifying (56) and u˜1,n,u˜2,n,u˜3,n is solution of (59) and (60) with R1,n,R2,n,R3,n,R4,n instead of h1,h2,h3,h4, and u˜1,n,u˜2,n,u˜3,n verifies (4). From (66), (69), and (56), it follows that w˜n and u˜1,n,u˜2,n,u˜3,n are bounded, respectively, in L2q3 and L2Ω3. Thus, there are sequences also denoted by w˜nn and u˜1,n,u˜2,n,u˜3,nn that converge weakly towards w˜ and u˜1,u˜2,u˜3, respectively, in L2q3 and L2Ω3. Moreover, w˜ verifies (56) and(83)η1=∫0Aβ3u˜3da,η2=∫0Aβ1u˜1da,η3=∫0Aσβ2u˜2da,η4=∫0A1−σβ2u˜2da.
Here,u˜1,u˜2,u˜3 is solution of (59) and (60) with R1,R2,R3,R4 instead of h1,h2,h3,h4, and u˜1,u˜2,u˜3 verifies (4). Therefore,(84)η1,η2,η3,η4∈ΛγR1,R2,R3,R4.
In addition, from (77) and Lions–Aubin lemma, one can extract a sequence ηn1,ηn2,ηn3,ηn4n that converges strongly to η1,η2,η3,η4 in ℳ. Since K is closed, then η1,η2,η3,η4∈K.
Finally, from (84), we conclude that R1,R2,R3,R4∈Λγ−1K.Proof.
of Theorem1.
This result is a consequence of Proposition2. In fact, from Proposition 2, there is R1,R2,R3,R4∈ℳ such that R1,R2,R3,R4∈ΛγR1,R2,R3,R4, see Theorem 1.3 in [22].


## 5. Numerical Simulations

In this section, we display the numerical simulations of problems (1) and (2). The first part is to reduce the PDE to a finite dimensional system of the form: X˙=AX+BY, where A and B are matrices.We construct the control problem, which consists in minimizing the functional and we choose the classical Hum functional and the control matrix B=χΘ, where Θ=0,A×0,T.Example 1.
For the simulation, we takek=0.25,A1=2,A2=3,A3=A4=4, and Δa=1/20. Moreover, the transition functions β1,β2, and β3 are given by(85)β1a=0,ifa=0,711a−44e−0.5210a−4Γ5,ifa≠0,β2a=0,ifa=0,750a−44e−0.5210a−4Γ5,ifa≠0,β3a=0,ifa=0,712.106a−43e−0.5210a−4Γ5,ifa≠0,and the initial population distributions ui0a=kie−1/Ai−a,1≤i≤4,withk1=0.5,k2=k4=1andk3=0.2 will be used in the computation. The growth rates are given by(86)v1a=0.2,ifa=0,a,ifa≠0,v2a=0.3,ifa=0,a,ifa≠0,v3a=0.4,ifa=0,a,ifa≠0,v4a=0.4,ifa=0,a,ifa≠0.
The following numerical results were obtained.
We notice that we have the positivity of the initial conditions, see Figure1. Moreover, the colors green, blue, and yellow show that the uncontrolled solutions are not null at the final time T=0.5 and the color red shows that the uncontrolled solutions are null at the final age A=4, see Figure 2.
We use ODE 45 for the simulations of the uncontrolled system.Figure 1
Initial conditions.Figure 2
Uncontrolled solution at final timeT=0.5 and A=4.Example 2.
In this example, we are going back to the data from Example1.
The control states and the controlled states results are obtained forT=0.5 and A=4.
We notice that numerically the control covers all the domains. However, we could not take a positivity constraint in our simulations, see Figure3.
Moreover, the colors red and yellow show that the controlled states are zero at the final timeT=0.5, see Figure 4. This confirms the result of null controllability obtained theoretically.
The CaSadi toolbox is used to simulate the control (by the minimization of functional: Hum Method).
Below, we have the numerical simulations results.Figure 3
Control state at final timeT=0.5 and A=4.Figure 4
Controlled solution at final timeT=0.5 and A=4.


## 6. Conclusion

In this paper, we have proved the null controllability of a four-stage and age-structured population dynamics model. We consider three controls acting on eggs, larvae, and females. We end with numerical illustrations to confirm the theoretical results. In order to consolidate the result obtained in this article, we are going to think about studying the controllability with less controls (two or one) and also to establish algorithms allowing to calculate the control from its characterization.



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