# Controllability of a Family of Nonlinear Population Dynamics Models

Authors: Yacouba Simporé
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operatorA and an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a time T dependent on the constraints on the age and the observability minimal time T0 of the pair A,B (B is the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..


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## Body

## 1. Introduction and Main Results

In this paper, we study the null controllability of an infinite-dimensional nonlinear system describing the dynamics of age-structured population.LetV,H, and U be a four separable Hilbert space. We identify H and U with their duals. We suppose V is dense in H and that the following inclusions are checked:(1)V⊆H⊆V′,where the inclusion of V in H is continuous.Now, we consider the operatorA is defined by(2)A:V⟶V′,where V is the Hilbert space and V⊆H⊆V′, for example, H1Ω⊆L2Ω⊆H−1Ω.And, we suppose that(3)A:V⟶V′,∀ϕ,ψ∈V⟶Aϕ,ψV′,V,is measurable. Let y be a solution of the following system:(4)∂y∂t+∂y∂a−Ay+μ0y=χa1,a2Bu,in0,A×0,T,y0,t=∫0AβaydaΦ∫0Aβayda,in0,T,ya,0=y0a,in0,A,where Φ∈L∞ℝ, and y0 is given in(5)K=L20,A;HandB∈ℒL20,T;U,L20,T;Ha bounded operator.Moreover, the positive functionμ0 denotes the natural mortality rate of individuals of age a, supposed to be independent of the times t.The control function isu, depending on t and a, where m is the characteristic function.We denote byβ the positive function describing the fertility rate that depends only on the age. The birth law ensures the survival of the species. In several works, one denotes ∫0Aβyda, the number of newborn individuals at time t. If we consider an oviparous species, that is to say whose birth process goes through egg laying, we see that ∫0Aβyda is the number of eggs laid at time t. Since all the eggs do not reach maturity, we introduce a function giving the proportion of eggs who will arrive there. Let Φ this function. Then, we have(6)y0,t=∫0AβydaΦ∫0Aβyda,the distribution of newborn individuals at time t.We assume that the fertility rateβ and the mortality rate μ0 satisfy the demographic property:(7)H1=i−μ0a≥0,for everya∈0,A,ii−μ0∈L10,A∗,for everyA∗∈0,A,iii−∫0Aμada=+∞,H2=i−β∈L∞0,A,ii−βa=0,∀a∈0,a^wherea^∈0,A.We define the probability of survival of an individual of age given by(8)πa=e−∫0aμ0sds.Thus, with hypothesisH1−iii, the probability of survival at maximal age A is zero. Moreover, hypothesis H2−ii means that individuals of age below a^ are not fertile. This hypothesis, in addition to being a demographic assumption, is also very important in establishing the observability inequality.For more details about the modelling of such system and the biological significance of the hypotheses, we refer to Webb [1].Moreover, there exist constantsα>0, γ>0, and λ such that, for every ϕ, ψ∈V, we have(9)H3=Aϕ,ψV′,V≤γϕVψV,−Aϕ,ϕV′,V+λϕH2≥αϕV2,H4:βμ0∈L10,A.In the following, we recall that the operatorA is independent of t and a and generator of a C0 semigroup on H.AssumptionsH3 and H4 are necessary for the existence of the fixed point. Indeed, they make it possible to establish an adequate regularity on the system for the existence of a fixed point.Theorem 1.
Lety0∈L20,A;H and ℬu∈L2a1,a2×0,T;U; under assumptions H1, H2, and H3, system (4) admits a unique solution y∈L20,T×0,A;H.Proof.
For the proof of the existence, we consider the following auxiliary system:(10)∂y1∂t+∂y1∂a−Ay1+μ0y1=e−λ0tℬu,in0,A×0,T,y10,t=∫0AβazdaΦ∫0Aβaeλ0tzda,in0,T,y1a,0=y0a,in0,A,where z∈L20,T;H and y1=e−λ0ty.
Using Theorem 2.8 of [2], the auxiliary system (10) admits a unique solution y1∈C0,A;L20,T;H.
Now, we consider the operatorΛ defined in L20,A×0,T;H on L20,A×0,T;H by Λz=y1z. Let y1z1 and y1z2 be the solutions of (10), where z is replaced, respectively, by z1andz2. We denote by Y=y1z1−y1z2. The function Y verifies(11)∂Y∂t+∂Y∂a−AY+μ0+λ0Y=0,in0,A×0,T,Y0,t=∫0Aβaz1daΦ∫0Aβaeλ0tz1da−∫0Aβaz2daΦ∫0Aβaeλ0tz2da,in0,T,Ya,0=y0a,in0,A.
By calculating the scalar product of the first equation of (11) by Y and integrating with respect to the age a and the time t, we obtain(12)∫0TYT,H2da+∫0AY.,AH2dt−∫0T∫0AAY,YV′,Vdadt+∫0T∫0Aμa+λ0YH2dadt=∫0T∫0Aβaz1daΦ∫0Aβaeλ0tz1da−∫0Aβaz2daΦ∫0Aβaeλ0tz2daH2dt.
Then,(13)∫0TYT,H2da+∫0AY.,AH2dt+λ0−λ∫0T∫0AYH2dadt≤M2Aβ∞2z1−z2L20,A×0,T;H2.
Choosingλ0 such that λ0−λ=2M2Aβ∞2 (λ is given in assumption H3), we obtain(14)YL20,A×0,T;H2≤12z1−z2L20,A×0,T;H2.
Therefore, the operator(15)Λz1−Λz2L20,A×0,T;H2≤12z1−z2L20,A×0,T;H2.
We conclude that, by the Banach fixed point thatΛ admits a fixed point, we have(16)Λy1=y1.
Finally, replacingz by y1=e−λ0ty in system (10), we obtain that y is solution of system (4).
Before we state our main result, let us introduce the notion of null controllability of the pairA,B. We have the following result.Definition 1.
We say that a pairA,B is null controllable in time T0 if, for every z0∈H, there exists a control u∈L20,T;U such that the solution of the system(17)z′t=Azt+But,t∈0,T,z0=z0,satisfies zT=0, for every T>T0.
The main result of this paper is as follows.Theorem 2.
Assume thatβ and μ satisfy conditions H1−H2−H3 above.
Assume thatA,B is null controllable in any time T>T0, with(18)T0<T^,T^=mina^−a1,a2−a1and a1<a^. Then, for every T>a1+A−a2+2T0 and for every y0∈L20,A;H, there exists a control u1∈L2a1,a2×0,T;U such that the solution y of (4) satisfies(19)ya,T=0,for alla∈0,A.This result can be seen as a generalization of those obtained by Traoré [3] and Simporé [4] in the case when A is an elliptic operator with Neumann or Dirichlet homogeneous boundary conditions and Echarroudi and Maniar [5] when A is a degenerate elliptic operator.Let us now mention some related works in controllability and observability from the literature. In the Network, Information, and Computer Security, Alcaraz and Wolthusen studied strategies for the efficient restoration of controllability following attacks and attacker-defender interactions in power-law networks in [6]. In [7], Alcaraz presents a network infrastructure based on three layers, where the redundant support is primarily concentrated on a fog-based structure to protect a specific subset of cyber-physical control devices.In the anticancer therapy, Swierniak and Kalmka study a local controllability of a model of combined anticancer therapy with delay in control (see [8]).For linear and nonlinear population dynamics models (Lotka–Mckendrick model), many results of local or global controllability have been established.Indeed, Ainseba and Iannelli, in [9], have worked on the controllability of two nonlinear dynamic model problems, the nonlinearity being on mortality and fertility. In the first model, the control corresponds to a supply of individuals on a small age interval. The second one is age and space structured, and the control corresponds to a supply of individuals on a small subdomain ω of the whole space domain Ω.In [3], Traoré proved the nonlinear Lotka–McKendrick is null controllable except for a small interval of ages near zero where the controls is localized with respect to the space variable but active for all ages.In [2], Maity et al. study the null controllability of the linear infinite-dimensional system obtained by grafting an age structure.To be in line with the studied models, many authors have also worked on constrained controllability problems. Indeed, the authors Maity et al., in [10], solved a problem of null controllability with positivity of the Lotka-Mckendrick system with spatial diffusion where the control is localized in the space variable as well as with respect to the age. The method combines final-state observability estimates with the use of characteristics and the associated semigroup.In [11], Kalmka worked in constrained approximate controllability IEEE Transactions on Automatic Control.In this paper, we extend the result of [2] to the nonlinear case. The nonlinearity is at the level of births and can be justified as follows.If we consider an oviparous species, that is to say whose birth process goes through egg-laying, we see that∫0Aβayda is the number of eggs laid at time t. Since all the eggs do not reach maturity, we introduce a function giving the proportion of eggs which will arrive there. Let Φ be this function. Then, we have,(20)Φ∫0Aβayda∫0Aβayda,the distribution of newborn individuals at time t.For the establishment of the observability inequality, we use the same technique as in [2, 10]. One of the advantages of this method is that the conditions ∫0Aμ0ada=+∞ on the mortality function do not interfere in establishing of the observability inequality which allows us to have a global controllability, which is not the case in [3, 9]. Moreover, we improve the observation domain, and better still we have null controllability for all ages. Indeed, in [3, 5, 9], the control is localized with respect to the space variable but active for all ages, and we do not have necessarily the extinction on the small interval of ages near zero. In this paper, the control is also localized with respect to the age variable, and we obtain the state of the system null for all ages.The remaining part of this work is organized as follows.In Section2, we establish the observability inequality of the adjoint system of an auxiliary linear system. This allows us to prove its approximate null controllability. Section 3 is devoted to the proof of the main result of this paper using a fixed point of Schauder, see for example [12]. To finish, we give some perspectives in the conclusion in Section 4.


## 2. Approximate Null Controllability of Auxiliary System

### 2.1. Adjoint System

We suppose that the pairA,B is final-state observable at time T0, and we consider the following auxiliary system (well chosen so as to obtain the main result of the paper by rotating a fixed point) given by(21)∂y∂t+∂y∂a−Ay+μ0ay=ℬu,in0,A×0,T,y0,t=G∫0Aβyda,inΩ×0,T,ya,0=y0,inΩ×0,A.Applying the scalar product of the first equation of (21) by q and integrating with respect to the age a and time t, we obtain the adjoint system of (21):(22)−∂q∂t−∂q∂a−A∗q+μ0aq−Gβq0,t=0,in0,A×0,T,qA,t=0,in0,T,qa,T=qTa,in0,A.For everyq0∈K, under assumption H1 and H2, system (22) admits a unique solution q. Moreover, integrating along the characteristic lines, the solution q of (22) is given by(23)qt=πa+T−tπaeT−tA∗qTa+T−t+∫tTπa+s−tπaes−tA∗Gsβa+s−tq0,sds,ifT−t≤A−a,∫tt+A−aπa+s−tπaes−tA∗Gsβa+s−tq0,sds,ifA−a<T−t,where πa=e−∫0aμ0αdα.We have the following result.Theorem 3.
Let us assume assumptionsH1−H2, and suppose the pair A,B in null controllable for every time T>T0. For T>A−a2+a1+2T0, there exists a control uɛ such that the solution y of system (21) verifies(24)ya,TL20,A;H≤ɛ,for everyɛ>0.For the proof of Theorem 1, we need the results of observability.


### 2.2. Observability Inequality

Theorem 4.
Under the assumption of Theorem3, for every qT∈K, the solution q of system (22) is final-state observable for every T>a1+A−a2+2T0. In other words, for every T>a1+A−a2+2T0, there exists KT>0 such that the solution q of (22) satisfies(25)qa,0L20,A;H2≤KTG∞,T,a1,a2,A∫0T∫a1a2B∗qa,tU2dxdadt.
For the proof, we need the following results.Proposition 1.
Let us assume assumptionH1−H2, and let a1<a^, T>T0+a1, and a1+T0<η<T; then, there exists a constant C>0 such that, for every qT∈K, the solution q of system (22) verifies the following inequality:(26)∫0T−ηq0,tH2dt≤C∫0T∫a1a2B∗qa,tU2dadt.Proposition 2.
Let us assume assumptionsH1−H2, and let a1<a^,T>T0+a1, and a1<a0<a2−T0; there exists CT>0 such that the solution q of system (26) verifies the following inequality:(27)qa,0L20,a0;H2≤CT∫0T∫a1a2B∗qa,tU2dadt.For the proofs of Propositions 1 and 2, we have the following proposition.Proposition 3.
Let us recall that the pairA∗,B∗ is final-state observable in any time T>T0 with T0≥0. Let CT be a observability cost with CT⟶+∞ as T⟶T0. Let T1,T2, and T3 be three real numbers such that 0≤T1≤T2≤T3 with T2−T1>T0. Then, for every w0∈DA∗, the solution w of the problem(28)∂wλ∂λ−A∗wλ=0,inT1,T3,wT1=w0,satisfies the estimate(29)wT3H2≤MT1,T2,T3∫T1T2ℬ∗wλU2dλ,where the constants c1 and c2 depend on T and Ω.Proof.
(see [2, 13]).
Fora∈0,a^, we have βa=0; therefore, system (22) can be written by(30)−∂q∂t−∂q∂a−A∗q+μ0q=0,in0,a^×0,T,qa,T=qTa,in0,a^.
We denote byq˜a,t=qa,te−∫0aμ0αdα. Then, q˜ satisfies(31)∂q˜∂t+∂q˜∂a−A∗q˜=0in0,a^×0,T.
Proving inequality (26) also leads to show that there exits a constant C>0 such that the solution q˜ of (31) satisfies(32)∫0T−ηq˜0,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.
Indeed, we have(33)∫0T−ηq0,tH2dt=∫0T−ηq˜0,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt≤C∫0T∫a1a2B∗qa,tU2dadt.Proof of Proposition 1.
We consider the following characteristics trajectoryγλ=T−λ,T+t−λ. If T−λ=0, the backward characteristics starting from 0,t. If T<a1, the trajectory γλ never reaches the observation region a1,a2 (see Figure 1). So, we choose T>a1.
Moreover, we estimateq0,t on 0,T−η, where η>a1. Indeed, all the characteristic starting on 0,t with t∈T−a1,T never intersects the observation domain (see Figure 1).
Without loss of the generality, let us assume hereη<a2≤a^<T.Figure 1
An illustration of the estimation ofqa,0 between 0 and a0 for case 1: green region corresponds to the interval 0,b0, black corresponds to the interval b0,a1, and blue corresponds to the interval a1,a0.


#### 2.2.1. Case 1: Fort∈0,T−a^

We denote by(34)wλ=q˜T−λ,T+t−λ;λ∈0,t.Then,w satisfies(35)∂wλ∂λ−A∗wλ=0,inT−a^,T,w0=q˜T,T+t.Using Proposition3 with 0<t0<t1<t, we obtain(36)wTH2≤wt1H2≤c1ec2/t1−t0∫t0t1B∗wλU2dλ,that is equivalent to(37)q˜0,tH2≤c1ec2/t1−t0∫t0t1B∗q˜T−λ,t+T−λ‵U2dλC∫T−t1T−t0B∗q˜a,t+aU2da.Then, fort0=T−a^ and t1=T−a1, we obtain(38)q˜20,tH2≤C∫a1a^B∗q˜2a,t+aU2da.Integrating with respect tot over 0,T−a^, we obtain(39)∫0T−a^q˜20,tH2dt≤C∫a1a^∫aT−a^+aB∗q˜2a,tU2dtda.Finally,(40)∫0T−a^q˜20,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.


#### 2.2.2. Case 2: Fort∈T−a^,T−η

We denote by(41)wλ=q˜T−λ,T+t−λ;λ∈T−a^,T.Then,w satisfies(42)∂wλ∂λ−A∗wλ,x=0,inT−a^,T,wT−a^=q˜a^,t+a^,inΩ.Using Proposition3 with T−a^<t0<t1<T, we obtain(43)wTH2≤wt1H2≤c1ec2/t1−t0∫t0t1B∗wλU2dλ,that is equivalent to(44)q˜0,tH2≤c1ec2/t1−t0∫t0t1B∗q˜T−λ,T+t−λUdλ.Then, fort0=T−η and t1=T−a1, we obtain(45)q˜0,tH2≤C∫a1ηB∗q˜a,t+aU2da.Integrating with respect tot over T−a^,T−η, we obtain(46)∫T−a^T−ηq˜0,tH2dt≤C∫a1η∫a+T−a^a+T−ηB∗q˜α,tU2dtda.Finally,(47)∫T−a^T−ηq˜0,H2dt≤C∫0T∫a1a2B∗q˜U2dadt.Combining (40) and (47), we obtain the result.Proving inequality (27) also leads to show that there exits a constant C>0 such that the solution q˜ of (31) satisfies(48)∫0a0q˜a,0H2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.Indeed, we have(49)∫0a0qa,0H2dt≤e2μL10,a0∫0a0q˜a,0H2dt≤Ce2μL10,a0∫0T∫a1a2B∗q˜a,tU2dadt≤Ce2μL10,a0∫0T∫a1a2B∗qa,tU2dadt.Proof of Proposition 1.
The proof will be done in two parts.


#### 2.2.3. The First Part: Interval0,a1

We consider in this proof the characteristicsγλ=a+λ,λ. For λ=0, the characteristics start from a,0.ForT>a1 and a∈0,a1, all the characteristics starting on a,0 enter the observation domain (see Figure 2). We have three cases.Case 1:T<a2 and a2≤a^.Two situations can arise:(i)
b0=a2−T<a1<a0: in this situation, we split the interval 0,a1 as(50)0,a1=0,b0∪b0,a1.Figure 2
An illustration of the estimate ofq0,t. Here, we have chosen a2=a^. Since t∈0,T−η with a1+T0<η<T, all the backward characteristics starting from 0,t enters the observation domain.(ii)
a1<b0<a0: in this situation, we keep the interval 0,a1.Case 2:T≥a2 and a2≤a^.In this case, we keep the interval0,a1.Case 3:a2>a^.In this case, we prove similarly the observability ina1,a^ and expand to a1,a2.Here, we give the proof in the only situation, where(51)T<a2,b0=a2−T<a1.In the remaining part of the proof, we give upper bounds for∫IB∗q˜a,0H2da, where I is successively each one of the intervals appearing in decomposition (50) (see Figure 2).Upper bound on0,b0: for a∈0,b0, we first set wλ=q˜T+a−λ,T−λλ∈0,Tandx∈Ω, where q˜=e−∫0aμ0αdαq.Then,w verifies(52)∂wλ∂λ−A∗wλ=0in0,T,wT=q˜a+T,T.By applying Proposition3 with t0=0 and t1=a+T−a1, we obtain(53)wTH2≤c1ec2/a+T−a1∫0a+T−a1B∗wλU2dλ.Then, we have(54)q˜a,0H2≤c1ec2/a+T−a1∫a1a+TB∗q˜α,α−aU2dλ=C∫a1a+Tℬ∗q˜α,α−aU2dα≤C∫a1a2B∗q˜α,α−aU2dα.Integrating with respecta over 0,b0, we obtain(55)∫0b0q˜a,0H2da≤C∫0b0∫a1a2B∗q˜α,α−aU2dαda.As(56)∫0b0∫a1a2ℬ∗q˜α,α−aU2dαda=∫a1a2∫0b0B∗q˜α,α−aU2dadα,then(57)∫0b0q˜a,0H2da≤C∫a1a2∫α−b0αB∗q˜α,tU2dtdα.Finally,(58)∫0b0q˜a,0H2da≤C∫0T∫a1a2B∗q˜a,tU2dadt.Upper boundb0,a1: for a∈b0,a1, we consider always system (52), but λ∈T+a−a2,T.Applying Proposition3 with t0=a+T−a2 and t1=a+T−a1, we obtain(59)q˜a,0H2≤C∫a1a2B∗q˜α,α−aU2dα.And as before, we obtain(60)∫b0a1q˜.0H2da≤C∫0T∫a1a2B∗q˜2U2dadt.


#### 2.2.4. The Second Part: Intervala1,a0

Here, we supposeT>T0 and a1<a0<a2−T0.And as in [2], we prove the existence of C>0 such that(61)∫a1a0q˜.0H2da≤C∫0T∫a1a2B∗q˜U2dadt.Consequently, combining (58), (60), and (61), we obtain(62)∫0a0q.0H2da≤C∫0T∫a1a2B∗qU2dadt.Lemma 1 (see Figure3).
Let us suppose thatT>A−a2+a1+2T0 with T0<T^. Then, there exists a0∈a1,mina^,a2−T0 and κ>0 such that(63)T−a1+T0+κ>A−a0.Figure 3
The backward characteristics starting froma,0 with a∈a2,A (green lines) hits the line a=A, gets renewed by the renewal condition βaq0,t, and then enters the observation domain. More precisely, in addition to the observation time T0 of the pair A,B, the backward characteristics need at most A−a2 time to hit the line a=A, get renewed by the renewal condition βaq0,t, and take maximum T0+a1 time to enter the observation domain. Thus, we need in all, at least, T=A−a2+a1+2T0 time to obtain the observability inequality. So, with the conditions T>A−a2+a1+2T0, T0>T^, and a1+T0<η<T, all the characteristics starting at a,0 with a∈a2,A get renewed by the renewal condition βaq0,t in t∈0,T−η and enter the observation domain.Proof.
Without loosing the generality, we suppose thata2=a^.
Suppose thatT>a1+A−a2+2T0⇔T−a1+T0>A−a2+T0. Then, there exists κ>0 (well chosen) such that T−a1−2κ−T0>A−a2+T0⇔T−a1+κ+T0>A−a2−κ−T0.
We denote bya0=a2−κ−T0; then, T−a1+κ+T0>A−a0.
We are in position to prove the observability inequality.Proof.
(proof of theorem 2). Leta0∈a1,mina^,a2−T0, as in the previous Lemma. We have(64)∫0Aqa,0H2da=∫0a0q.0H2da+∫a0Aq.0H2da.
From the result of (23) and applying the previous lemma, there exists a0∈0,a^ such that(65)qa,0=∫0A−aπ1a+sπ1aGsβa+sq0,sds,and then(66)∫a0Aqa,0H2da≤K2∫0T−a1+κ+T0q0,H2dt.
Finally, with the result of Proposition1, we obtain(67)∫a0Aq.0H2da≤K3∫0T∫a1a2B∗qU2dadt.
With the result of Proposition2 and inequality (67), we obtain the result of Theorem 4 (see Figure 3).
The figures below illustrate, respectively, the estimate of the nonlocal term, the estimate ofqa,0, and the observation time.Proof. of the Theorem 3.
Forɛ>0, we consider the functional Jɛ define by(68)Jɛu=12∫0T∫a1a2uU2dadt+12ɛ∫0Ay.TH2da,where y is the solution of the following system:(69)∂y∂t+∂y∂a−Ay+μ0y=ℬu,in0,A×0,T,y0,t=Gl∫0Aβayda,in0,T,ya,0=y0,in0,A,with l∈L20,T;H.Lemma 2.
The functionalJɛ is continuous, strictly convex, and coercive. Consequently, Jɛ reaches its minimum at a point uɛ∈L2a1,a2×0,T;U.
Moreover, settingyɛ, the associated solution of (69), and qɛ, the solution (21) with qɛa,T=1/ɛyɛa,T, has uɛ=ℬ∗qɛ, and there exist C1>0 and C2>0 independent of ɛ and l, such that(70)∫0T∫a1a2B∗qϵU2dadt≤C1y0L20,A;H2yɛ.TL20,A;H2≤ɛC2y0L20,A;H2.Proof.
(proof of lemma). It is easy to check thatJɛ is coercive, continuous, and strictly convex. Then, it admits a unique minimizer uɛ. The maximum principle gives that uɛ=ℬ∗qɛ.
By calculating the dot product of the first equation of (69) with u=ℬ∗qɛ by Y and integrating with respect to the age a and the time t, we obtain(71)∫0T∫a1a2B∗qϵU2dadt+1ɛ∫0Ayɛ2.TH2da=∫0Ay0,qɛ.0Hda.
Using the inequality of Young, we obtain, for anyδ>0,(72)∫0T∫a1a2B∗qɛU2dadt+1ɛ∫0Ayϵ2.TH2da≤12∫0Ay0H2da+12∫0Aqɛ.0H2da.
Using the observability inequality (25) and choosing KT=δ, where KT is given in (25), we obtain(73)12∫0T∫a1a2B∗qɛU2dadt+12ɛ∫0Ayɛ2.TH2da≤12∫0Ay0H2da.
Then, the result is obtained.


## 2.1. Adjoint System

We suppose that the pairA,B is final-state observable at time T0, and we consider the following auxiliary system (well chosen so as to obtain the main result of the paper by rotating a fixed point) given by(21)∂y∂t+∂y∂a−Ay+μ0ay=ℬu,in0,A×0,T,y0,t=G∫0Aβyda,inΩ×0,T,ya,0=y0,inΩ×0,A.Applying the scalar product of the first equation of (21) by q and integrating with respect to the age a and time t, we obtain the adjoint system of (21):(22)−∂q∂t−∂q∂a−A∗q+μ0aq−Gβq0,t=0,in0,A×0,T,qA,t=0,in0,T,qa,T=qTa,in0,A.For everyq0∈K, under assumption H1 and H2, system (22) admits a unique solution q. Moreover, integrating along the characteristic lines, the solution q of (22) is given by(23)qt=πa+T−tπaeT−tA∗qTa+T−t+∫tTπa+s−tπaes−tA∗Gsβa+s−tq0,sds,ifT−t≤A−a,∫tt+A−aπa+s−tπaes−tA∗Gsβa+s−tq0,sds,ifA−a<T−t,where πa=e−∫0aμ0αdα.We have the following result.Theorem 3.
Let us assume assumptionsH1−H2, and suppose the pair A,B in null controllable for every time T>T0. For T>A−a2+a1+2T0, there exists a control uɛ such that the solution y of system (21) verifies(24)ya,TL20,A;H≤ɛ,for everyɛ>0.For the proof of Theorem 1, we need the results of observability.


## 2.2. Observability Inequality

Theorem 4.
Under the assumption of Theorem3, for every qT∈K, the solution q of system (22) is final-state observable for every T>a1+A−a2+2T0. In other words, for every T>a1+A−a2+2T0, there exists KT>0 such that the solution q of (22) satisfies(25)qa,0L20,A;H2≤KTG∞,T,a1,a2,A∫0T∫a1a2B∗qa,tU2dxdadt.
For the proof, we need the following results.Proposition 1.
Let us assume assumptionH1−H2, and let a1<a^, T>T0+a1, and a1+T0<η<T; then, there exists a constant C>0 such that, for every qT∈K, the solution q of system (22) verifies the following inequality:(26)∫0T−ηq0,tH2dt≤C∫0T∫a1a2B∗qa,tU2dadt.Proposition 2.
Let us assume assumptionsH1−H2, and let a1<a^,T>T0+a1, and a1<a0<a2−T0; there exists CT>0 such that the solution q of system (26) verifies the following inequality:(27)qa,0L20,a0;H2≤CT∫0T∫a1a2B∗qa,tU2dadt.For the proofs of Propositions 1 and 2, we have the following proposition.Proposition 3.
Let us recall that the pairA∗,B∗ is final-state observable in any time T>T0 with T0≥0. Let CT be a observability cost with CT⟶+∞ as T⟶T0. Let T1,T2, and T3 be three real numbers such that 0≤T1≤T2≤T3 with T2−T1>T0. Then, for every w0∈DA∗, the solution w of the problem(28)∂wλ∂λ−A∗wλ=0,inT1,T3,wT1=w0,satisfies the estimate(29)wT3H2≤MT1,T2,T3∫T1T2ℬ∗wλU2dλ,where the constants c1 and c2 depend on T and Ω.Proof.
(see [2, 13]).
Fora∈0,a^, we have βa=0; therefore, system (22) can be written by(30)−∂q∂t−∂q∂a−A∗q+μ0q=0,in0,a^×0,T,qa,T=qTa,in0,a^.
We denote byq˜a,t=qa,te−∫0aμ0αdα. Then, q˜ satisfies(31)∂q˜∂t+∂q˜∂a−A∗q˜=0in0,a^×0,T.
Proving inequality (26) also leads to show that there exits a constant C>0 such that the solution q˜ of (31) satisfies(32)∫0T−ηq˜0,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.
Indeed, we have(33)∫0T−ηq0,tH2dt=∫0T−ηq˜0,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt≤C∫0T∫a1a2B∗qa,tU2dadt.Proof of Proposition 1.
We consider the following characteristics trajectoryγλ=T−λ,T+t−λ. If T−λ=0, the backward characteristics starting from 0,t. If T<a1, the trajectory γλ never reaches the observation region a1,a2 (see Figure 1). So, we choose T>a1.
Moreover, we estimateq0,t on 0,T−η, where η>a1. Indeed, all the characteristic starting on 0,t with t∈T−a1,T never intersects the observation domain (see Figure 1).
Without loss of the generality, let us assume hereη<a2≤a^<T.Figure 1
An illustration of the estimation ofqa,0 between 0 and a0 for case 1: green region corresponds to the interval 0,b0, black corresponds to the interval b0,a1, and blue corresponds to the interval a1,a0.


### 2.2.1. Case 1: Fort∈0,T−a^

We denote by(34)wλ=q˜T−λ,T+t−λ;λ∈0,t.Then,w satisfies(35)∂wλ∂λ−A∗wλ=0,inT−a^,T,w0=q˜T,T+t.Using Proposition3 with 0<t0<t1<t, we obtain(36)wTH2≤wt1H2≤c1ec2/t1−t0∫t0t1B∗wλU2dλ,that is equivalent to(37)q˜0,tH2≤c1ec2/t1−t0∫t0t1B∗q˜T−λ,t+T−λ‵U2dλC∫T−t1T−t0B∗q˜a,t+aU2da.Then, fort0=T−a^ and t1=T−a1, we obtain(38)q˜20,tH2≤C∫a1a^B∗q˜2a,t+aU2da.Integrating with respect tot over 0,T−a^, we obtain(39)∫0T−a^q˜20,tH2dt≤C∫a1a^∫aT−a^+aB∗q˜2a,tU2dtda.Finally,(40)∫0T−a^q˜20,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.


### 2.2.2. Case 2: Fort∈T−a^,T−η

We denote by(41)wλ=q˜T−λ,T+t−λ;λ∈T−a^,T.Then,w satisfies(42)∂wλ∂λ−A∗wλ,x=0,inT−a^,T,wT−a^=q˜a^,t+a^,inΩ.Using Proposition3 with T−a^<t0<t1<T, we obtain(43)wTH2≤wt1H2≤c1ec2/t1−t0∫t0t1B∗wλU2dλ,that is equivalent to(44)q˜0,tH2≤c1ec2/t1−t0∫t0t1B∗q˜T−λ,T+t−λUdλ.Then, fort0=T−η and t1=T−a1, we obtain(45)q˜0,tH2≤C∫a1ηB∗q˜a,t+aU2da.Integrating with respect tot over T−a^,T−η, we obtain(46)∫T−a^T−ηq˜0,tH2dt≤C∫a1η∫a+T−a^a+T−ηB∗q˜α,tU2dtda.Finally,(47)∫T−a^T−ηq˜0,H2dt≤C∫0T∫a1a2B∗q˜U2dadt.Combining (40) and (47), we obtain the result.Proving inequality (27) also leads to show that there exits a constant C>0 such that the solution q˜ of (31) satisfies(48)∫0a0q˜a,0H2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.Indeed, we have(49)∫0a0qa,0H2dt≤e2μL10,a0∫0a0q˜a,0H2dt≤Ce2μL10,a0∫0T∫a1a2B∗q˜a,tU2dadt≤Ce2μL10,a0∫0T∫a1a2B∗qa,tU2dadt.Proof of Proposition 1.
The proof will be done in two parts.


### 2.2.3. The First Part: Interval0,a1

We consider in this proof the characteristicsγλ=a+λ,λ. For λ=0, the characteristics start from a,0.ForT>a1 and a∈0,a1, all the characteristics starting on a,0 enter the observation domain (see Figure 2). We have three cases.Case 1:T<a2 and a2≤a^.Two situations can arise:(i)
b0=a2−T<a1<a0: in this situation, we split the interval 0,a1 as(50)0,a1=0,b0∪b0,a1.Figure 2
An illustration of the estimate ofq0,t. Here, we have chosen a2=a^. Since t∈0,T−η with a1+T0<η<T, all the backward characteristics starting from 0,t enters the observation domain.(ii)
a1<b0<a0: in this situation, we keep the interval 0,a1.Case 2:T≥a2 and a2≤a^.In this case, we keep the interval0,a1.Case 3:a2>a^.In this case, we prove similarly the observability ina1,a^ and expand to a1,a2.Here, we give the proof in the only situation, where(51)T<a2,b0=a2−T<a1.In the remaining part of the proof, we give upper bounds for∫IB∗q˜a,0H2da, where I is successively each one of the intervals appearing in decomposition (50) (see Figure 2).Upper bound on0,b0: for a∈0,b0, we first set wλ=q˜T+a−λ,T−λλ∈0,Tandx∈Ω, where q˜=e−∫0aμ0αdαq.Then,w verifies(52)∂wλ∂λ−A∗wλ=0in0,T,wT=q˜a+T,T.By applying Proposition3 with t0=0 and t1=a+T−a1, we obtain(53)wTH2≤c1ec2/a+T−a1∫0a+T−a1B∗wλU2dλ.Then, we have(54)q˜a,0H2≤c1ec2/a+T−a1∫a1a+TB∗q˜α,α−aU2dλ=C∫a1a+Tℬ∗q˜α,α−aU2dα≤C∫a1a2B∗q˜α,α−aU2dα.Integrating with respecta over 0,b0, we obtain(55)∫0b0q˜a,0H2da≤C∫0b0∫a1a2B∗q˜α,α−aU2dαda.As(56)∫0b0∫a1a2ℬ∗q˜α,α−aU2dαda=∫a1a2∫0b0B∗q˜α,α−aU2dadα,then(57)∫0b0q˜a,0H2da≤C∫a1a2∫α−b0αB∗q˜α,tU2dtdα.Finally,(58)∫0b0q˜a,0H2da≤C∫0T∫a1a2B∗q˜a,tU2dadt.Upper boundb0,a1: for a∈b0,a1, we consider always system (52), but λ∈T+a−a2,T.Applying Proposition3 with t0=a+T−a2 and t1=a+T−a1, we obtain(59)q˜a,0H2≤C∫a1a2B∗q˜α,α−aU2dα.And as before, we obtain(60)∫b0a1q˜.0H2da≤C∫0T∫a1a2B∗q˜2U2dadt.


### 2.2.4. The Second Part: Intervala1,a0

Here, we supposeT>T0 and a1<a0<a2−T0.And as in [2], we prove the existence of C>0 such that(61)∫a1a0q˜.0H2da≤C∫0T∫a1a2B∗q˜U2dadt.Consequently, combining (58), (60), and (61), we obtain(62)∫0a0q.0H2da≤C∫0T∫a1a2B∗qU2dadt.Lemma 1 (see Figure3).
Let us suppose thatT>A−a2+a1+2T0 with T0<T^. Then, there exists a0∈a1,mina^,a2−T0 and κ>0 such that(63)T−a1+T0+κ>A−a0.Figure 3
The backward characteristics starting froma,0 with a∈a2,A (green lines) hits the line a=A, gets renewed by the renewal condition βaq0,t, and then enters the observation domain. More precisely, in addition to the observation time T0 of the pair A,B, the backward characteristics need at most A−a2 time to hit the line a=A, get renewed by the renewal condition βaq0,t, and take maximum T0+a1 time to enter the observation domain. Thus, we need in all, at least, T=A−a2+a1+2T0 time to obtain the observability inequality. So, with the conditions T>A−a2+a1+2T0, T0>T^, and a1+T0<η<T, all the characteristics starting at a,0 with a∈a2,A get renewed by the renewal condition βaq0,t in t∈0,T−η and enter the observation domain.Proof.
Without loosing the generality, we suppose thata2=a^.
Suppose thatT>a1+A−a2+2T0⇔T−a1+T0>A−a2+T0. Then, there exists κ>0 (well chosen) such that T−a1−2κ−T0>A−a2+T0⇔T−a1+κ+T0>A−a2−κ−T0.
We denote bya0=a2−κ−T0; then, T−a1+κ+T0>A−a0.
We are in position to prove the observability inequality.Proof.
(proof of theorem 2). Leta0∈a1,mina^,a2−T0, as in the previous Lemma. We have(64)∫0Aqa,0H2da=∫0a0q.0H2da+∫a0Aq.0H2da.
From the result of (23) and applying the previous lemma, there exists a0∈0,a^ such that(65)qa,0=∫0A−aπ1a+sπ1aGsβa+sq0,sds,and then(66)∫a0Aqa,0H2da≤K2∫0T−a1+κ+T0q0,H2dt.
Finally, with the result of Proposition1, we obtain(67)∫a0Aq.0H2da≤K3∫0T∫a1a2B∗qU2dadt.
With the result of Proposition2 and inequality (67), we obtain the result of Theorem 4 (see Figure 3).
The figures below illustrate, respectively, the estimate of the nonlocal term, the estimate ofqa,0, and the observation time.Proof. of the Theorem 3.
Forɛ>0, we consider the functional Jɛ define by(68)Jɛu=12∫0T∫a1a2uU2dadt+12ɛ∫0Ay.TH2da,where y is the solution of the following system:(69)∂y∂t+∂y∂a−Ay+μ0y=ℬu,in0,A×0,T,y0,t=Gl∫0Aβayda,in0,T,ya,0=y0,in0,A,with l∈L20,T;H.Lemma 2.
The functionalJɛ is continuous, strictly convex, and coercive. Consequently, Jɛ reaches its minimum at a point uɛ∈L2a1,a2×0,T;U.
Moreover, settingyɛ, the associated solution of (69), and qɛ, the solution (21) with qɛa,T=1/ɛyɛa,T, has uɛ=ℬ∗qɛ, and there exist C1>0 and C2>0 independent of ɛ and l, such that(70)∫0T∫a1a2B∗qϵU2dadt≤C1y0L20,A;H2yɛ.TL20,A;H2≤ɛC2y0L20,A;H2.Proof.
(proof of lemma). It is easy to check thatJɛ is coercive, continuous, and strictly convex. Then, it admits a unique minimizer uɛ. The maximum principle gives that uɛ=ℬ∗qɛ.
By calculating the dot product of the first equation of (69) with u=ℬ∗qɛ by Y and integrating with respect to the age a and the time t, we obtain(71)∫0T∫a1a2B∗qϵU2dadt+1ɛ∫0Ayɛ2.TH2da=∫0Ay0,qɛ.0Hda.
Using the inequality of Young, we obtain, for anyδ>0,(72)∫0T∫a1a2B∗qɛU2dadt+1ɛ∫0Ayϵ2.TH2da≤12∫0Ay0H2da+12∫0Aqɛ.0H2da.
Using the observability inequality (25) and choosing KT=δ, where KT is given in (25), we obtain(73)12∫0T∫a1a2B∗qɛU2dadt+12ɛ∫0Ayɛ2.TH2da≤12∫0Ay0H2da.
Then, the result is obtained.


## 2.2.1. Case 1: Fort∈0,T−a^

We denote by(34)wλ=q˜T−λ,T+t−λ;λ∈0,t.Then,w satisfies(35)∂wλ∂λ−A∗wλ=0,inT−a^,T,w0=q˜T,T+t.Using Proposition3 with 0<t0<t1<t, we obtain(36)wTH2≤wt1H2≤c1ec2/t1−t0∫t0t1B∗wλU2dλ,that is equivalent to(37)q˜0,tH2≤c1ec2/t1−t0∫t0t1B∗q˜T−λ,t+T−λ‵U2dλC∫T−t1T−t0B∗q˜a,t+aU2da.Then, fort0=T−a^ and t1=T−a1, we obtain(38)q˜20,tH2≤C∫a1a^B∗q˜2a,t+aU2da.Integrating with respect tot over 0,T−a^, we obtain(39)∫0T−a^q˜20,tH2dt≤C∫a1a^∫aT−a^+aB∗q˜2a,tU2dtda.Finally,(40)∫0T−a^q˜20,tH2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.


## 2.2.2. Case 2: Fort∈T−a^,T−η

We denote by(41)wλ=q˜T−λ,T+t−λ;λ∈T−a^,T.Then,w satisfies(42)∂wλ∂λ−A∗wλ,x=0,inT−a^,T,wT−a^=q˜a^,t+a^,inΩ.Using Proposition3 with T−a^<t0<t1<T, we obtain(43)wTH2≤wt1H2≤c1ec2/t1−t0∫t0t1B∗wλU2dλ,that is equivalent to(44)q˜0,tH2≤c1ec2/t1−t0∫t0t1B∗q˜T−λ,T+t−λUdλ.Then, fort0=T−η and t1=T−a1, we obtain(45)q˜0,tH2≤C∫a1ηB∗q˜a,t+aU2da.Integrating with respect tot over T−a^,T−η, we obtain(46)∫T−a^T−ηq˜0,tH2dt≤C∫a1η∫a+T−a^a+T−ηB∗q˜α,tU2dtda.Finally,(47)∫T−a^T−ηq˜0,H2dt≤C∫0T∫a1a2B∗q˜U2dadt.Combining (40) and (47), we obtain the result.Proving inequality (27) also leads to show that there exits a constant C>0 such that the solution q˜ of (31) satisfies(48)∫0a0q˜a,0H2dt≤C∫0T∫a1a2B∗q˜a,tU2dadt.Indeed, we have(49)∫0a0qa,0H2dt≤e2μL10,a0∫0a0q˜a,0H2dt≤Ce2μL10,a0∫0T∫a1a2B∗q˜a,tU2dadt≤Ce2μL10,a0∫0T∫a1a2B∗qa,tU2dadt.Proof of Proposition 1.
The proof will be done in two parts.


## 2.2.3. The First Part: Interval0,a1

We consider in this proof the characteristicsγλ=a+λ,λ. For λ=0, the characteristics start from a,0.ForT>a1 and a∈0,a1, all the characteristics starting on a,0 enter the observation domain (see Figure 2). We have three cases.Case 1:T<a2 and a2≤a^.Two situations can arise:(i)
b0=a2−T<a1<a0: in this situation, we split the interval 0,a1 as(50)0,a1=0,b0∪b0,a1.Figure 2
An illustration of the estimate ofq0,t. Here, we have chosen a2=a^. Since t∈0,T−η with a1+T0<η<T, all the backward characteristics starting from 0,t enters the observation domain.(ii)
a1<b0<a0: in this situation, we keep the interval 0,a1.Case 2:T≥a2 and a2≤a^.In this case, we keep the interval0,a1.Case 3:a2>a^.In this case, we prove similarly the observability ina1,a^ and expand to a1,a2.Here, we give the proof in the only situation, where(51)T<a2,b0=a2−T<a1.In the remaining part of the proof, we give upper bounds for∫IB∗q˜a,0H2da, where I is successively each one of the intervals appearing in decomposition (50) (see Figure 2).Upper bound on0,b0: for a∈0,b0, we first set wλ=q˜T+a−λ,T−λλ∈0,Tandx∈Ω, where q˜=e−∫0aμ0αdαq.Then,w verifies(52)∂wλ∂λ−A∗wλ=0in0,T,wT=q˜a+T,T.By applying Proposition3 with t0=0 and t1=a+T−a1, we obtain(53)wTH2≤c1ec2/a+T−a1∫0a+T−a1B∗wλU2dλ.Then, we have(54)q˜a,0H2≤c1ec2/a+T−a1∫a1a+TB∗q˜α,α−aU2dλ=C∫a1a+Tℬ∗q˜α,α−aU2dα≤C∫a1a2B∗q˜α,α−aU2dα.Integrating with respecta over 0,b0, we obtain(55)∫0b0q˜a,0H2da≤C∫0b0∫a1a2B∗q˜α,α−aU2dαda.As(56)∫0b0∫a1a2ℬ∗q˜α,α−aU2dαda=∫a1a2∫0b0B∗q˜α,α−aU2dadα,then(57)∫0b0q˜a,0H2da≤C∫a1a2∫α−b0αB∗q˜α,tU2dtdα.Finally,(58)∫0b0q˜a,0H2da≤C∫0T∫a1a2B∗q˜a,tU2dadt.Upper boundb0,a1: for a∈b0,a1, we consider always system (52), but λ∈T+a−a2,T.Applying Proposition3 with t0=a+T−a2 and t1=a+T−a1, we obtain(59)q˜a,0H2≤C∫a1a2B∗q˜α,α−aU2dα.And as before, we obtain(60)∫b0a1q˜.0H2da≤C∫0T∫a1a2B∗q˜2U2dadt.


## 2.2.4. The Second Part: Intervala1,a0

Here, we supposeT>T0 and a1<a0<a2−T0.And as in [2], we prove the existence of C>0 such that(61)∫a1a0q˜.0H2da≤C∫0T∫a1a2B∗q˜U2dadt.Consequently, combining (58), (60), and (61), we obtain(62)∫0a0q.0H2da≤C∫0T∫a1a2B∗qU2dadt.Lemma 1 (see Figure3).
Let us suppose thatT>A−a2+a1+2T0 with T0<T^. Then, there exists a0∈a1,mina^,a2−T0 and κ>0 such that(63)T−a1+T0+κ>A−a0.Figure 3
The backward characteristics starting froma,0 with a∈a2,A (green lines) hits the line a=A, gets renewed by the renewal condition βaq0,t, and then enters the observation domain. More precisely, in addition to the observation time T0 of the pair A,B, the backward characteristics need at most A−a2 time to hit the line a=A, get renewed by the renewal condition βaq0,t, and take maximum T0+a1 time to enter the observation domain. Thus, we need in all, at least, T=A−a2+a1+2T0 time to obtain the observability inequality. So, with the conditions T>A−a2+a1+2T0, T0>T^, and a1+T0<η<T, all the characteristics starting at a,0 with a∈a2,A get renewed by the renewal condition βaq0,t in t∈0,T−η and enter the observation domain.Proof.
Without loosing the generality, we suppose thata2=a^.
Suppose thatT>a1+A−a2+2T0⇔T−a1+T0>A−a2+T0. Then, there exists κ>0 (well chosen) such that T−a1−2κ−T0>A−a2+T0⇔T−a1+κ+T0>A−a2−κ−T0.
We denote bya0=a2−κ−T0; then, T−a1+κ+T0>A−a0.
We are in position to prove the observability inequality.Proof.
(proof of theorem 2). Leta0∈a1,mina^,a2−T0, as in the previous Lemma. We have(64)∫0Aqa,0H2da=∫0a0q.0H2da+∫a0Aq.0H2da.
From the result of (23) and applying the previous lemma, there exists a0∈0,a^ such that(65)qa,0=∫0A−aπ1a+sπ1aGsβa+sq0,sds,and then(66)∫a0Aqa,0H2da≤K2∫0T−a1+κ+T0q0,H2dt.
Finally, with the result of Proposition1, we obtain(67)∫a0Aq.0H2da≤K3∫0T∫a1a2B∗qU2dadt.
With the result of Proposition2 and inequality (67), we obtain the result of Theorem 4 (see Figure 3).
The figures below illustrate, respectively, the estimate of the nonlocal term, the estimate ofqa,0, and the observation time.Proof. of the Theorem 3.
Forɛ>0, we consider the functional Jɛ define by(68)Jɛu=12∫0T∫a1a2uU2dadt+12ɛ∫0Ay.TH2da,where y is the solution of the following system:(69)∂y∂t+∂y∂a−Ay+μ0y=ℬu,in0,A×0,T,y0,t=Gl∫0Aβayda,in0,T,ya,0=y0,in0,A,with l∈L20,T;H.Lemma 2.
The functionalJɛ is continuous, strictly convex, and coercive. Consequently, Jɛ reaches its minimum at a point uɛ∈L2a1,a2×0,T;U.
Moreover, settingyɛ, the associated solution of (69), and qɛ, the solution (21) with qɛa,T=1/ɛyɛa,T, has uɛ=ℬ∗qɛ, and there exist C1>0 and C2>0 independent of ɛ and l, such that(70)∫0T∫a1a2B∗qϵU2dadt≤C1y0L20,A;H2yɛ.TL20,A;H2≤ɛC2y0L20,A;H2.Proof.
(proof of lemma). It is easy to check thatJɛ is coercive, continuous, and strictly convex. Then, it admits a unique minimizer uɛ. The maximum principle gives that uɛ=ℬ∗qɛ.
By calculating the dot product of the first equation of (69) with u=ℬ∗qɛ by Y and integrating with respect to the age a and the time t, we obtain(71)∫0T∫a1a2B∗qϵU2dadt+1ɛ∫0Ayɛ2.TH2da=∫0Ay0,qɛ.0Hda.
Using the inequality of Young, we obtain, for anyδ>0,(72)∫0T∫a1a2B∗qɛU2dadt+1ɛ∫0Ayϵ2.TH2da≤12∫0Ay0H2da+12∫0Aqɛ.0H2da.
Using the observability inequality (25) and choosing KT=δ, where KT is given in (25), we obtain(73)12∫0T∫a1a2B∗qɛU2dadt+12ɛ∫0Ayɛ2.TH2da≤12∫0Ay0H2da.
Then, the result is obtained.


## 3. Null Controllability of Nonlinear System

Now, we consider the following system:(74)∂yl∂t+∂yl∂a−Ayl+μ0yl=χa1,a2BB∗q,in0,A×0,T,yl0,t=Gl∫0Aβaylda,in0,T,yla,0=y0,in0,A,where Gl=Φl,l∈L20,T;H.We have the following.Lemma 3.
Under assumptionH3, there exists C>0 and K>0 such that the solution yɛl of (69) verifies(75)yɛlL20,A×0,T;H2+∫0T∫0Aμ0yɛlH2dadt≤Ky0L20,A;H2yɛl.AL20,T;H2≤Cy0L20,A;H2.Proof.
(proof of the lemma). For the proof, we denote bypɛl=yɛleλ0t. Then, pɛl satisfies(76)∂pɛl∂t+∂pɛl∂a−Apɛl+μ0pɛl=1a1,a2e−λ0tBB∗qɛl,in0,A×0,T,pɛl0,t=Gl∫0Aβpɛlda,in0,T,pɛla,0=y0a,in0,A.
The functionpɛl verifies(77)∂pɛl∂t+∂pɛl∂a+Apɛl+λ0+μ0pɛl=1a1,a2e−λ0tBB∗qɛl.
By calculating the dot product of the equation of (77) by pɛl and integrating with respect the age a and the times t, we obtain(78)∫0ApɛlT,H2da+∫0Tpɛl.AH2dt−∫0T∫0AApɛl,pɛlV′,Vdadt+∫0T∫0Aλ0+μ0pɛlH2dadt=∫0T∫0Apɛl,1a1,a2e−λ0tBB∗qɛlH,Udadt+∫0Apɛl0,H2da+∫0Tpɛl.0H2dt.
We have(79)∫0T∫0AB∗pɛl,1a1,a2e−λ0tB∗qɛlU,Udadt≤12δ∫0T∫0AB∗pɛlU2dadt+δ2∫0T∫a1a2B∗qϵlU2.
As(80)∫0T∫0AB∗pɛl,1a1,a2e−λ0tB∗qɛlU,Udadt≤C12δ∫0T∫0ApɛlH2dadt+δ2∫0T∫a1a2B∗qɛlU2,moreover(81)∫0Tpɛl.0H2dt≤C2β∞,ApɛlL20,T×0,A;H2.
Using assumptionH3 and choosing λ0 and δ such that(82)λ0=C12δ+C2β∞,A+γ+1,we obtain(83)pɛ.AL20,T2+∫0T∫0Aμ0pɛlH2dadt+αpɛlL20,T;V2+pɛlL20,T×0,A;V2≤Cy0L20,A;H2,(with α and λ the parameters given in the assumption H3).
Therefore, we have the result.
LetΛ be an operator defined on L20,T;H by(84)Λl=∫0Aβyɛlda,where yɛl is the solution of the following system.Proposition 4.
We suppose that operatorA verifies the hypotheses H3. Then, the operator Λ is continuous, bounded, and compact of that L20,T;H. Therefore, Λ admits a fixed point.Proof.
(proof of the proposition 3). Let(85)M=∫0Aβyɛlda.
Then,Y=e−λ0tM is the solution of the following system:(86)∂Yl∂t−AYl+λ0Yl=e−λ0t∫a1a2βBB∗qϵda+R,in0,A×0,T,Yl.0=∫0Aβy0da,in0,T,where R=∫0Aμ0βeλ0tyɛlda+∫0Aβ′yɛlda−βAyɛl.A.
From the results of the previous lemma, we have(87)yɛlL20,A×0,T;H≤Ky0L20,T;H.
By calculating the dot product of the first equation of (87) by Yl and integrating with respect the age a and the times t, we obtain(88)Yl.TH2−∫0TAYl,YlV′,Vdt+λ0∫0TYlH2dt=∫0TYl,∫a1a2e−λ0tBB∗qɛlH,Udt+∫0TR,YlH,Hdt+Yl.0H2.
We have, using Young inequality,(89)∫0TB∗Yl,∫a1a2e−λ0tB∗qɛlU,Udt+∫0TR,YlH,Hdt≤δ2∫0TB∗YlU2dt+12δ∫0T∫a1a2B∗qɛlU2dadt+12δ∫0TRH2dt+δ2∫0TYlH2dt.
Then,(90)∫0TB∗Yl,∫a1a2e−λ0tB∗qɛlU,Udt+∫0TR,YlH,Hdt≤δC+12∫0TYlH2dt+12δ∫0T∫a1a2B∗qɛlU2dadt+12δ∫0TRH2dt.
Boundness ofR: we have(91)RL20,T:H2≤2eλ0T∫0T∫0Aμ0βyɛldaH2dt+2β′∞yɛlL20,A×0,T;H2+2βAyɛl.A,L20,T;H2.
Asμ0β∈L10,1, using Cauchy Schwarz inequality, we obtain(92)RL20,T:H2≤2eλ0T∫0Aμ0βda∫0T∫0Aμ0βμ0βyɛldaH2dadt+2β′∞yɛlL20,A×0,T;H2+2βAyɛl.A,L20,T;H2.
Then,(93)RL20,T:H2≤2eλ0T∫0Aμ0βda∫0T∫0Aμ0βyɛlH2dadt+2β′∞yɛlL20,A×0,T;H2+2βAyɛl.A,L20,T;H2.
Finally, in Lemma3, there exists KT>0 such that(94)RL20,T:H2≤KTy0L20,T;H2.
Now, using assumptionH3 and choosing λ0=δC+1/2+1+λ, we obtain(95)YlTH2+αYlL20,T;V2+YlL20,T,H2≤Ky0L20,A×0,T;H2,(with α and λ the parameters given in the assumption H3).
Then,M is bounded in L20,T;V independently of l and ɛ.
Proceeding in the same way by calculating the dot product of the first equation of (86) by a test function ψ, we obtain(96)∫0T∂Y∂t,ψdt≤CTψL20,T;Vy0L20,A;H.
Then,(97)∂Y∂tL20,T;V′≤CTy0L20,A;H.
Then,M is bounded in L20,T;V and ∂M/∂t is bounded in L20,T;V′ independently of l and ɛ.
Hence, using the Lions–Aubin lemma, we conclude thatΛ is bounded and compact in L20,T;H.


### 3.1. Step 2: Continuity of the OperatorΛ

Letm1,ɛp and m2,ɛp be the solution of the following cascade system:(98)∂m1,ɛl∂t+∂m1,ɛl∂a−Am1,ɛl+μ0m1,ɛl=χa1,a2B∗vɛ,in0,A×0,T,m1,ɛla,0=0,in0,A,m1,ɛl0,t=Φl∫0Aβm1,ɛlda,in0,T,(99)∂m0,ɛl∂t+∂m0,ɛl∂a−Am0,ɛl+μ0m0,ɛl=0,in0,A×0,T,m0,ɛla,0=m0,in0,A,m0,ɛl0,t=Φl∫0Aβam0,ϵlda,in0,T.Due to Lemma2, there exists vɛ∈L2a1,a2×0,T;U such that(100)m1,ɛl.T+m0,ɛl.T≤κ,for any ɛ>0.Any suitable control can be chosen, in particular, the control of minimum norm inL2a1,a2×0,T;U. Let us then consider(101)yɛl=m1,ɛl+m0,ɛl,where v1,ɛ is the optimal control characterised by Lemma 2 such that (100) holds, and let us consider the operator:(102)Λ′:L20,T;H⟶L20,A×0,T;H,l⟶yɛl.The functionΦl is bounded, when l spans L20,T;H. Since the solution m1,ɛ on system (33) depends continuously on the data v1,ɛ, it follows that the operator Λ′ is continuous. Moreover, the operator Λ″l′=∫0Aβal′da is continuous on L20,T;H, and therefore, we conclude that the operator Λ is continuous.Schauder’s fixed-point theorem (see [4, 12]) implies that Λ admits a fixed point. Then, there exists l∈H such that(103)Λl=l=∫0Aβyɛda,and yɛ solves(104)∂yɛ∂t+∂yɛ∂a−Ayɛ+μ0ayɛ=1a,a2BB∗qɛ,in0,A×0,T,yɛ0,t=∫0AβyɛdaΦ∫0Aβyɛda,in0,T,yɛa,0=y0a,in0,A.Moreover, we have(105)∫0T∫a1a2B∗qɛU2dadt≤C1y0L20,A;H2,yɛ.TL20,A;H2≤ɛC2y0L20,A;H2.Finally, whenϵ⟶0,(106)B∗qɛ⇀u,yɛT,H⟶yT,H=0.


## 3.1. Step 2: Continuity of the OperatorΛ

Letm1,ɛp and m2,ɛp be the solution of the following cascade system:(98)∂m1,ɛl∂t+∂m1,ɛl∂a−Am1,ɛl+μ0m1,ɛl=χa1,a2B∗vɛ,in0,A×0,T,m1,ɛla,0=0,in0,A,m1,ɛl0,t=Φl∫0Aβm1,ɛlda,in0,T,(99)∂m0,ɛl∂t+∂m0,ɛl∂a−Am0,ɛl+μ0m0,ɛl=0,in0,A×0,T,m0,ɛla,0=m0,in0,A,m0,ɛl0,t=Φl∫0Aβam0,ϵlda,in0,T.Due to Lemma2, there exists vɛ∈L2a1,a2×0,T;U such that(100)m1,ɛl.T+m0,ɛl.T≤κ,for any ɛ>0.Any suitable control can be chosen, in particular, the control of minimum norm inL2a1,a2×0,T;U. Let us then consider(101)yɛl=m1,ɛl+m0,ɛl,where v1,ɛ is the optimal control characterised by Lemma 2 such that (100) holds, and let us consider the operator:(102)Λ′:L20,T;H⟶L20,A×0,T;H,l⟶yɛl.The functionΦl is bounded, when l spans L20,T;H. Since the solution m1,ɛ on system (33) depends continuously on the data v1,ɛ, it follows that the operator Λ′ is continuous. Moreover, the operator Λ″l′=∫0Aβal′da is continuous on L20,T;H, and therefore, we conclude that the operator Λ is continuous.Schauder’s fixed-point theorem (see [4, 12]) implies that Λ admits a fixed point. Then, there exists l∈H such that(103)Λl=l=∫0Aβyɛda,and yɛ solves(104)∂yɛ∂t+∂yɛ∂a−Ayɛ+μ0ayɛ=1a,a2BB∗qɛ,in0,A×0,T,yɛ0,t=∫0AβyɛdaΦ∫0Aβyɛda,in0,T,yɛa,0=y0a,in0,A.Moreover, we have(105)∫0T∫a1a2B∗qɛU2dadt≤C1y0L20,A;H2,yɛ.TL20,A;H2≤ɛC2y0L20,A;H2.Finally, whenϵ⟶0,(106)B∗qɛ⇀u,yɛT,H⟶yT,H=0.


## 4. Applications

AssumptionsH3 and H4 are not necessary in the case of the linear system.


### 4.1. Nonlinear Population Dynamics System Structuring by Age and Spatial Diffusion

We consider the following nonlinear population dynamics structuring by age and spatial position:(107)∂y∂t+∂y∂a−Ay+μ0ay=1a1,a2Bu,inQ=Ω×0,A×0,T,Boundary condition,onΣ=∂Ω×0,A×0,T,yx,0,s,t=Φ∫0Aβyda∫0Aβyda,inΩ×0,T,yx,a,s,0=y0,inΩ×0,A,where A is a operator define in the domain Ω⊂RN, N∈ℕ, and B=1ω.For(108)Aψ=∑i,j=1n∂∂xiσi,j∂ψ∂xj,withσij=σji∈C2Ω¯for1≤i,j≤N,we assume there exists a constant C>0 such that(109)∑i,j=1Nσi,jxξiξj>Cξ2x∈Ω¯,ξ∈ℝN,DA=ψ∈H2Ωsuch thatAψ∈L2Ωand∂ψ∂νA=∑i=1Nσi,j∂ψ∂xi⋅ni=0.The pairA,B is observable for every time T>0. Moreover, the operator A verifies hypotheses H3, where V=H1Ω,H=L2Ω. Then, applying Theorem 2, the system (107) is null controllable for every time T>a1+A−a2.


### 4.2. Nonlinear Population Dynamics Models with Degenerate Diffusion

We consider the following nonlinear population dynamics with degenerate diffusion (see [5]):(110)∂y∂t+∂y∂a−∂xkx∂xy+μ0ay=1a1,a2Bu,inQ=Ω×0,A×0,T,y0,a,t=0orayx0,a,t=0,in0,A×0,T,y1,a,t=0,in0,A×0,T,yx,0,s,t=Φ∫0Aβyda∫0Aβyda,inΩ×0,T,yx,a,s,0=y0,inΩ×0,A,where k is the nonnegative function in Ω=0,1 and degenerates. Moreover, B=χl1,l2.Let us set the state and the control space as follows:(111)V=Hk1Ω=φ∈L2Ωabsolutely continuous in0,1∫01kx∂xφ2dx<+∞,φ0=φ1=0,Hk2Ω=φ∈Hk1Ωkφx∈H1Ω,equipped with the following norm(112)φHk12=φL2Ω2+k∂xφL2Ω2,φHk22=φHk1Ω2+∂xk∂xφL2Ω2.We consider the unbounded operator A onH=L2Ω defined by(113)DA=Hk2Ω,Aψ=∂xkx∂xψ.Proposition 5 (see [14].
The operatorA:DA⟶L2Ω is closed, self-adjoint, and negative with dense domain.
Hence,A is the infinitesimal generator of a strongly continuous semigroup etA on L2Ω.The functionk∈C00,1∩C10,1 is such that it satisfies k0=0 and k>0 in Ω, and there exists M1∈0,1 such that(114)xk′x≤M1kx,for allx∈0,1.Under the above assumptions, the pairA,B is observable for every time T>0 (see [15]).Moreover, the operatorA verifies hypotheses H3, where V=Hk1ΩandH=L2Ω.Now, multiplying∂xkx∂xφ by ψ and integrating by part over 0,1, where φandψ∈Hk1Ω, we obtain(115)∫01∂xkx∂xφψdx=∫01kxφxψxdx.Using the Cauchy–Schwarz inequality, we obtain(116)∫01∂xkx∂xφψdx≤Ckx∂xφL2Ωkx∂xψL2Ω.Then,(117)∫01∂xkx∂xφψdx≤CφHk1ΩψHk1Ω.Similarly, we have(118)−∫01∂xkx∂xφφdx=∫01kx∂xφ2dx.Then, for allγ≥1 and α<1, we have(119)−∫01∂xkx∂xφψdx+γ∫01φ2dx=∫01kx∂xφ2dx+γ∫01φ2dx≥αφHk1Ω2.Then, applying Theorem2, system (110) is null controllable for every time T>a1+A−a2.


### 4.3. Nonlinear Population Dynamics Models with Degenerate Diffusion (the Nondivergence Form)

We consider the following nonlinear population dynamics with degenerate diffusion (see [5]):(120)∂y∂t+∂y∂a−kx∂xx2y+μ0ay=1a1,a2Bu,inQ=Ω×0,A×0,T,y0,a,t=y1,a,t=0,0,A×0,T,yx,0,s,t=Φ∫0Aβyda∫0Aβyda,inΩ×0,T,yx,a,s,0=y0,inΩ×0,A,where k is the nonnegative function in Ω=0,1 and degenerates. Moreover, B=χl1,l2.Let us set the state and the control space as follows:(121)H=L1/k2Ω=φ∈L2Ω|∫01φ2kdx<+∞,V=H1/k1Ω=L1/k2Ω∩H01Ω,H1/k2Ω=φ∈H1/k1Ω|kφxx2∈L1/k2Ω,with the following norms:(122)φHk12=1kφL2Ω2+∂xφL2Ω2,φHk22=φHk1Ω2+k∂xx2φL2Ω2.We consider the unbounded operator A onH=L1/k2Ω defined by(123)DA=H1/k2Ω,Aψ=kx∂xx2ψ.Proposition 6 (see [16]).
The operatorA:DA⟶L2Ω is closed, self-adjoint, and negative with dense domain.Hence,A is the infinitesimal generator of a strongly continuous semigroup etA on L1/k2Ω.The functionk∈C00,1∩C30,1 is such thatThe functionxkx/k∈L∞0,ɛ, and there exists M1∈0,2 and R1>0 such that xkx/k≤M1 and ∂xxxkx/k≤R11/kx for all x∈0,ϵ.The functionx−1kx/k∈L∞1−ɛ,1, and there exists M2∈0,2 and R2>0 such that x−1kx/k≤M2 and ∂xxx−1kx/k≤C1/kx for all x∈1−ɛ,1.Under the above assumptions, the pairA,B is observable for every time T>0 (see [16]).Moreover, the operatorA verifies assumption H3, where V=H1/k1ΩandH=L1/k2Ω.Indeed, letφandψ∈H1/k1Ω, and we have(124)kx∂xx2φ,ψH1/k1′,H1/k1=∫01φxψxdx.Using the Cauchy–Schwarz inequality, we obtain(125)kx∂xx2φ,ψH1/k1′,H1/k1≤∂xφL2Ω∂xψL2Ω.Using the Hardy–Poincaré inequality,(126)for allφ∈H1/k1Ω,∃C>0such that∫01φ2kdx≤C∫01φx2dx,we obtain(127)kx∂xx2φ,ψH1/k1′,H1/k1≤φH1/k1ΩψH1/k1Ω.Similarly, we have(128)−kx∂xx2φ,φH1/k1′,H1/k1=∫01∂xφ2dx.Then, for allγ≥1 and α<1, we have(129)−kx∂xx2φ,φH1/k1′,H1/k1+γ∫01φ2kdx=∫01∂xφ2dx+γ∫01φ2kdx≥αφH1/k1Ω2.Then, applying Theorem2, we obtain this result.Proposition 7.
By applying Theorem2, for y0∈L20,T;L1/k2Ω and for every T>a1+A−a2, there exists u∈L20,A×0,T;L1/k2Ω such that the solution of system (120) verify(130)y.T=0,a.e inΩ×0,A.


## 4.1. Nonlinear Population Dynamics System Structuring by Age and Spatial Diffusion

We consider the following nonlinear population dynamics structuring by age and spatial position:(107)∂y∂t+∂y∂a−Ay+μ0ay=1a1,a2Bu,inQ=Ω×0,A×0,T,Boundary condition,onΣ=∂Ω×0,A×0,T,yx,0,s,t=Φ∫0Aβyda∫0Aβyda,inΩ×0,T,yx,a,s,0=y0,inΩ×0,A,where A is a operator define in the domain Ω⊂RN, N∈ℕ, and B=1ω.For(108)Aψ=∑i,j=1n∂∂xiσi,j∂ψ∂xj,withσij=σji∈C2Ω¯for1≤i,j≤N,we assume there exists a constant C>0 such that(109)∑i,j=1Nσi,jxξiξj>Cξ2x∈Ω¯,ξ∈ℝN,DA=ψ∈H2Ωsuch thatAψ∈L2Ωand∂ψ∂νA=∑i=1Nσi,j∂ψ∂xi⋅ni=0.The pairA,B is observable for every time T>0. Moreover, the operator A verifies hypotheses H3, where V=H1Ω,H=L2Ω. Then, applying Theorem 2, the system (107) is null controllable for every time T>a1+A−a2.


## 4.2. Nonlinear Population Dynamics Models with Degenerate Diffusion

We consider the following nonlinear population dynamics with degenerate diffusion (see [5]):(110)∂y∂t+∂y∂a−∂xkx∂xy+μ0ay=1a1,a2Bu,inQ=Ω×0,A×0,T,y0,a,t=0orayx0,a,t=0,in0,A×0,T,y1,a,t=0,in0,A×0,T,yx,0,s,t=Φ∫0Aβyda∫0Aβyda,inΩ×0,T,yx,a,s,0=y0,inΩ×0,A,where k is the nonnegative function in Ω=0,1 and degenerates. Moreover, B=χl1,l2.Let us set the state and the control space as follows:(111)V=Hk1Ω=φ∈L2Ωabsolutely continuous in0,1∫01kx∂xφ2dx<+∞,φ0=φ1=0,Hk2Ω=φ∈Hk1Ωkφx∈H1Ω,equipped with the following norm(112)φHk12=φL2Ω2+k∂xφL2Ω2,φHk22=φHk1Ω2+∂xk∂xφL2Ω2.We consider the unbounded operator A onH=L2Ω defined by(113)DA=Hk2Ω,Aψ=∂xkx∂xψ.Proposition 5 (see [14].
The operatorA:DA⟶L2Ω is closed, self-adjoint, and negative with dense domain.
Hence,A is the infinitesimal generator of a strongly continuous semigroup etA on L2Ω.The functionk∈C00,1∩C10,1 is such that it satisfies k0=0 and k>0 in Ω, and there exists M1∈0,1 such that(114)xk′x≤M1kx,for allx∈0,1.Under the above assumptions, the pairA,B is observable for every time T>0 (see [15]).Moreover, the operatorA verifies hypotheses H3, where V=Hk1ΩandH=L2Ω.Now, multiplying∂xkx∂xφ by ψ and integrating by part over 0,1, where φandψ∈Hk1Ω, we obtain(115)∫01∂xkx∂xφψdx=∫01kxφxψxdx.Using the Cauchy–Schwarz inequality, we obtain(116)∫01∂xkx∂xφψdx≤Ckx∂xφL2Ωkx∂xψL2Ω.Then,(117)∫01∂xkx∂xφψdx≤CφHk1ΩψHk1Ω.Similarly, we have(118)−∫01∂xkx∂xφφdx=∫01kx∂xφ2dx.Then, for allγ≥1 and α<1, we have(119)−∫01∂xkx∂xφψdx+γ∫01φ2dx=∫01kx∂xφ2dx+γ∫01φ2dx≥αφHk1Ω2.Then, applying Theorem2, system (110) is null controllable for every time T>a1+A−a2.


## 4.3. Nonlinear Population Dynamics Models with Degenerate Diffusion (the Nondivergence Form)

We consider the following nonlinear population dynamics with degenerate diffusion (see [5]):(120)∂y∂t+∂y∂a−kx∂xx2y+μ0ay=1a1,a2Bu,inQ=Ω×0,A×0,T,y0,a,t=y1,a,t=0,0,A×0,T,yx,0,s,t=Φ∫0Aβyda∫0Aβyda,inΩ×0,T,yx,a,s,0=y0,inΩ×0,A,where k is the nonnegative function in Ω=0,1 and degenerates. Moreover, B=χl1,l2.Let us set the state and the control space as follows:(121)H=L1/k2Ω=φ∈L2Ω|∫01φ2kdx<+∞,V=H1/k1Ω=L1/k2Ω∩H01Ω,H1/k2Ω=φ∈H1/k1Ω|kφxx2∈L1/k2Ω,with the following norms:(122)φHk12=1kφL2Ω2+∂xφL2Ω2,φHk22=φHk1Ω2+k∂xx2φL2Ω2.We consider the unbounded operator A onH=L1/k2Ω defined by(123)DA=H1/k2Ω,Aψ=kx∂xx2ψ.Proposition 6 (see [16]).
The operatorA:DA⟶L2Ω is closed, self-adjoint, and negative with dense domain.Hence,A is the infinitesimal generator of a strongly continuous semigroup etA on L1/k2Ω.The functionk∈C00,1∩C30,1 is such thatThe functionxkx/k∈L∞0,ɛ, and there exists M1∈0,2 and R1>0 such that xkx/k≤M1 and ∂xxxkx/k≤R11/kx for all x∈0,ϵ.The functionx−1kx/k∈L∞1−ɛ,1, and there exists M2∈0,2 and R2>0 such that x−1kx/k≤M2 and ∂xxx−1kx/k≤C1/kx for all x∈1−ɛ,1.Under the above assumptions, the pairA,B is observable for every time T>0 (see [16]).Moreover, the operatorA verifies assumption H3, where V=H1/k1ΩandH=L1/k2Ω.Indeed, letφandψ∈H1/k1Ω, and we have(124)kx∂xx2φ,ψH1/k1′,H1/k1=∫01φxψxdx.Using the Cauchy–Schwarz inequality, we obtain(125)kx∂xx2φ,ψH1/k1′,H1/k1≤∂xφL2Ω∂xψL2Ω.Using the Hardy–Poincaré inequality,(126)for allφ∈H1/k1Ω,∃C>0such that∫01φ2kdx≤C∫01φx2dx,we obtain(127)kx∂xx2φ,ψH1/k1′,H1/k1≤φH1/k1ΩψH1/k1Ω.Similarly, we have(128)−kx∂xx2φ,φH1/k1′,H1/k1=∫01∂xφ2dx.Then, for allγ≥1 and α<1, we have(129)−kx∂xx2φ,φH1/k1′,H1/k1+γ∫01φ2kdx=∫01∂xφ2dx+γ∫01φ2kdx≥αφH1/k1Ω2.Then, applying Theorem2, we obtain this result.Proposition 7.
By applying Theorem2, for y0∈L20,T;L1/k2Ω and for every T>a1+A−a2, there exists u∈L20,A×0,T;L1/k2Ω such that the solution of system (120) verify(130)y.T=0,a.e inΩ×0,A.


## 5. Conclusion

Considering a nonlinear infinite dimensional system obtained by grafting an operatorA and an age structure, we show there is a time T dependent on the constraints on the age and the observability minimal time T′ of the pair A,B (B is the control operator) from which the system is null controllable. In application, the spatial diffusion operator and two spatial degenerate diffusion operators are provided.In the desire to consolidate even more the result obtained in this article, we will think in our future work of the case of controllability with positivity constraint and also to establish algorithms allowing to calculate the control from its characterization.These outstanding questions are therefore in this order:Controllability with positivity constraints: we will be interested in controllability with positivity constraint.Numerical implementation: for a given fertility rateβ, the mortality rate `μ, the initial condition y0, and a positive parameter ɛ>0, how to determine a numerical algorithm allowing to determine the ϵ-approximate null control function h?



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