# Null Controllability of a Nonlinear Dissipative System and Application to the Detection of the Incomplete Parameter for a Nonlinear Population Dynamics Model

Authors: Yacouba Simporé; Oumar Traoré
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

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## Abstract

We first prove a null controllability result for a nonlinear system derived from a nonlinear population dynamics model. In order to tackle the controllability problem we use an adapted Carleman inequality. Next we consider the nonlinear population dynamics model with a source term called the pollution term. In order to obtain information on the pollution term we use the method of sentinel.


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## Body

## 1. Introduction and Main Result

For a given real functionG, in this paper we consider the following nonlinear system: (1)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xG∫0Aβqda+χΘh+χωvinQ=0,T×0,A×Ω,qt,a,x=0onΣqt,A,x=0in0,T×0,AqT,a,x=0in0,A×Ω,where Ω is a bounded open subset of RN, N≥1, with a smooth boundary ∂Ω. Let T be a positive real and let ω, Θ be two open subsets of Ω such that ω¯⊂Ω and Θ⊂Ω. Here χω, χΘ are the characteristic function of ω and Θ, Δ is the Laplacian with respect to the spatial variable, and βa and μa denote, respectively, the natural fertility and the natural death rate of individuals of age a.The purpose of this paper is to prove a null controllability result for (1) at any time T. This means more precisely that we prove here the existence of a control v∈L2Qω such that the associated solution of (1) verifies q0,a,x=0 a.e. in 0,A×Ω.For the sequel we assume that the following assumptions hold:H1μa≥0lima→A∫0aμsds=+∞,μ∈Lloc⁡10,A,H2β∈C20,Aβa≥0in0,A∃0<a0<a1<Asuch thatβa=0in0,a0∪a1,AH3G∈L∞R.We set Qω=0,T×0,A×ω and QΘ=0,T×0,A×Θ.Let us now present the main results of this paper.For this aim we introduce the weight(2)eMt,a=exp⁡MatA-aT-tand define the Hilbert space(3)L2eM=f∈L2Q:∫Qf2eMdxdadt<∞,endowed with the natural norm. M is a real that will be defined in the following.We have the following.Theorem 1.
Leth∈L2eM; then under the assumptions H1, H2, and H3 there exists a control v∈L2Qω satisfying q0,a,x=0 a.e. in 0,A×Ω.The rest of this paper is as follows: the proof of the main result is given in Section2; in Section 3, we describe the method of sentinel; and Section 4 is devoted to the information provided by the sentinel.


## 2. Proof of the Null Controllability of the Nonlinear System

### 2.1. An Adapted Observability Inequality

We recall here that there exists a functionΨ∈C2Ω¯ such that Ψx=0, for all x∈∂Ω; Ψx>0 for all x∈Ω and ∇Ψ≠0 for all Ω-ω~¯, where ω~ is an open set such that ω~¯⊂ω⊂Ω. See [1].Now we consider the following system:(4)∂w∂t+∂w∂a-Δw+μw=finQw=0onΣw0,a,x=w0a,xin0,A×Ωwt,0,x=gt,xin0,T×Ω.Proposition 2.
There exist positive constantsα1>1 and λ1>1 and there exists a positive constant CA,T which depends on A and T such that, for α≥α1, λ≥λ1, and for all solution of (4) the following inequality holds: (5)α3λ4∫Qe-2αηp3w2dxdadt≤C∫Qe-2αηf2dxdadt+Cα3λ4∫Qωe-2αηp3w2dxdadtwith(6)ηt,a,x=e2λΨ∞-eλΨxatT-tA-a,pt,a,x=eλΨxatT-tA-a.For the proof of this proposition see [2].Forf=0 we obtain an Observability Inequality.Lemma 3.
There existM>0 and C>0 such that the following inequality is true:(7)∫Qexp⁡-MatT-tA-aw2dxdadt≤C∫Qωw2dxdadt;herein w is the solution of system (4).Proof.
Iff=0, (5) become(8)∫Qe-2αηp3w2dxdadt≤C∫Qωe-2αηp3w2dxdadt.Choosing λ=λ1 and α=α0≥α1 such that e-2α0ηp3≤1, we obtain the result with M=2α0exp⁡(2λ1Ψ∞).In the following we consider the linear system:(9)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xbt,x+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×0,AqT,a,x=0in0,A×Ω,(10)∂φ∂t+∂φ∂a-Δφ+μφ=0inQφ=0onΣφ0,a,x=φ0in0,A×Ωφt,0,x=∫0Aβφbt,xdain0,T×Ω,where at,x and bt,x are in L∞0,T×Ω. Each system (9)-(10) is well posed and admits a unique solution. See [3] for the proof of existence and uniqueness of the solution.Remark 4.
For the linear case, that is,Gx=x can use the Hahn-Banach theorem for proving the existence of a control v which verifies q0,·,·L20,A×Ω≤ϵ, one writes(11)R0=q0,a,x;v∈L2Qωwhenqsolves9.It is easy to see that R0 is a nonempty subspace of L20,A×Ω and that (12)R⊥0is  dense  inL20,A×Ω.It suffices to show that (13)R⊥0⊂0.Suppose that (14)φ0∈R⊥0.Multiplying (9) by φ and integrating by parts, we have(15)∫QχΘh+χωvφdxdadt=∫0A∫Ωq0,a,xφ0a,xdadx.Since (16)φ0∈R⊥0,then (17)∫0A∫Ωq0,a,xφ0dadx=0.This implies that (18)∫QχΘh+χωvφdxdadt=0for every v∈L2Qω.Thenφ=0 on Θ∩ω×0,T×0,A. The unique continuation property of φ gives (19)φ=0on0,T×0,A×Ω.Therefore φ0=0. So R⊥0⊂0.However, this method cannot allow the study of the nonlinear case.


### 2.2. The Approached Controllability of the Linear System

Now, forδ>0 we consider the functional Jδ defined by (20)Jδv=12∫Qωv2dxdadt+1δ∫0A∫Ωq20,a,xdxda,where q solves(21)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xbt,x+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×ΩqT,a,x=0in0,A×Ω.Proposition 5.
The functionalJδ is continuous, strictly convex, and coercive. Therefore, Jδ reaches its minimum at a point vδ∈L2Qω. Moreover, setting qδ the associated solution of (21) and φδ the solution of (10) with φδ0,a,x=-(1/δ)qδ0,a,x has χωvδ=χωφδ and there exist C>0, K0>0 independent of δ, such that(22)∫qωφδ2dxdadt≤ChL2eM2,∫0A∫Ωqδ20,a,xdxda≤δK0hL2eM2.Remark 6.
As we will see Proposition5 solves the problem of existence of the approached sentinel for the linear case.Proof.
It is easy to check thatJδ is coercive, continuous, and strictly convex. Then it admits a unique minimizer vδ. The maximum principle gives that χωvδ=χωφδ.
Multiplying (21) with v=vδ by φδ and integrating by parts, we obtain (23)-1δ∫0A∫Ωqδ20,a,xdxda=∫QΘhφδdxdadt+∫Qωvδφδdxdadt.So (24)1δ∫0A∫Ωqδ20,a,xdxda+∫Qωφδ2dtdadx≤∫QΘhφδdxdadt.Using the inequality of Young, we obtain for any R>0(25)1δ∫0A∫Ωqδ20,a,xdxda+∫Qωφδ2dtdadx≤2RhL2eM2+12R∫Qexp⁡-MatT-tA-aφδ2dxdadt.Now, inequalities (7) and (25) give(26)1δ∫0A∫Ωqδ20,a,xdxda+∫Qωφδ2dxdadt≤2RhL2eM2+C2R∫Qωφδ2dxdadt.Choosing R such that 2R=C we have(27)1δ∫0A∫Ωqδ20,a,xdxda+12∫Qωφδ2dxdadt≤2RhL2eM2.We conclude that (28)∫Qωφδ2dtdadx≤2RhL2eM2,∫0A∫Ωqδ20,a,xdxda≤2RδhL2eM2.Proposition 7.
The functionqδ solution of system (21) verifies (29)qδL20,T×0,A;H1Ω≤ChL2eM+hL2Q.Proof.
Letλ0 be a positive constant. Put q^δ=e-λ0tqδ.
Then,q^δ solves(30)-∂q^δ∂t-∂q^δ∂a-Δq^δ+μ+λ0q^δ=βq^δt,0,xbt,x+χΘe-λ0th+χωe-λ0tφδinQq^δt,a,x=0onΣq^δt,A,x=0in0,T×Ωq^δT,a,x=0in0,A×Ω.Multiplying (30) by q^δ and integrating by parts, we obtain(31)12∫0A∫Ωq^δ20,a,xdxda+12∫0T∫Ωq^δ2t,0,xdxdt+∫Q∇q^δ2dxdadt+λ0q^δL2Q2≤12∫0T∫Ωq^δ2t,0,xdxdt+Aβ∞2b∞2q^δL2Q2+hL2Q2+Cq^δL2Q2+2RhL2eM2.Choosing λ0>Aβ∞2b∞2+C+1, we obtain (32)∇q^δL2Q+q^δL2Q≤ChL2Q+hL2eM.This gives the desired estimation.


### 2.3. Study of the Nonlinear Case

In the following we set(33)bt,x=Gη.Let Λ be an operator defined on L20,T×Ω by (34)Λη=∫0Aβaqδηda,where qδη is the solution of the following system:(35)-∂qδ∂t-∂qδ∂a-Δqδ+μqδ=βqδt,0,xGη+χΘh+χωφδinQqδt,a,x=0onΣqδt,A,x=0in0,T×0,AqδT,a,x=0in0,A×Ω.Then we have the following.Proposition 8.
The operatorΛ is continuous, bounded, and compact of L20,T×Ω. Then Λ admits a fixed point.Proof.
The proof will be done in two steps as follows.
Step  1 (boundedness and compactness of
Λ
). Let (36)Yt,a,x=∫0Aβaqδda,and denote 0,T×Ω=QT.
ThenY is the solution of the following system:(37)∂Yη∂t-ΔYη+∫0Aμβaqδda=ZηinQTYηt,x=0on0,T×∂ΩYη0,x=0inΩwith(38)Zη=∫0Aβa′qδ+χωβφδ+β2qδt,0,xGη+βhda.Under assumptions H1, H2, and H3 and the results and owing to the estimation on qδ and φδ the function Zη satisfies (39)ZηL20,T×Ω≤Cβa′∞,β∞,G∞hL2eM+hL2Q.System (37) is a heat equation where the source term and the initial condition are bounded, respectively, in L2QT and L2Ω.
ThenYη is bounded in L20,T;H01Ω and ∂Yη/∂t is bounded in L20,T;H0-1Ω. Hence, using the Lions-Aubin lemma we conclude that Λ is bounded and compact in L2QT.
Step  2 (continuity of
Λ
). Let ηn→η strongly in L2QT. Then we can show that the sequence Yηn converges to Yη strongly. For all ηn∈R, qδηn and φδηn are bounded independently to ηn. Therefore Zηn is bounded in L2QT. Then we can extract a subsequence Xηnk such that(40)Yηk⟶YηonQTZηk⟶Zηweakly  inQT∫0Aμβqδηnda⟶∫0Aμβqδηdaweakly  inL2QT.Then Yη is a solution of (37). We deduce that the sequence Yηn converges to Yη.
Since the operatorΛ is continuous, bounded, and compact on L2QT onto L2QT, Schauder’s fixed-point theorem (see [4, 5]) implies that Λ admits a fixed point.
There existsη∈L2QT such that (41)Λη=η=∫0Aβqδda.Then qδ solves(42)-∂qδ∂t-∂qδ∂a-Δqδ+μqδ=βqδt,0,xG∫0Aβqδda+χΘh+χωφδinQqδt,a,x=0onΣqδt,A,x=0in0,T×0,AqδT,a,x=0in0,A×Ω.From the foregoing, we have that φδ and qδ are bounded in L2Q.
Then we can extract a subsequence ofqδ still denoted by qδn, such that (43)qδ⇀qinL2Q,∫0Aβqδda⟶∫0AβqdainL2QT. By the same idea (44)χωφδ⇀χωvinL2Q.Since ∫0Aβqδda→∫0Aβqda, then there exists a subsequence still denoted by ∫0Aβqδda such that ∫0Aβqδda→∫0Aβqda a.e in QT.
Now sinceG is continuous, then(45)G∫0Aβqδda⟶G∫0Aβqdaa.e  inQT.Therefore, one derives that q solves the following system:(46)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xG∫0Aβqda+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×0,AqT,a,x=0in0,A×Ω,and we have also, for δ→0, q0,a,x=0 a.e. in 0,A×Ω.


## 2.1. An Adapted Observability Inequality

We recall here that there exists a functionΨ∈C2Ω¯ such that Ψx=0, for all x∈∂Ω; Ψx>0 for all x∈Ω and ∇Ψ≠0 for all Ω-ω~¯, where ω~ is an open set such that ω~¯⊂ω⊂Ω. See [1].Now we consider the following system:(4)∂w∂t+∂w∂a-Δw+μw=finQw=0onΣw0,a,x=w0a,xin0,A×Ωwt,0,x=gt,xin0,T×Ω.Proposition 2.
There exist positive constantsα1>1 and λ1>1 and there exists a positive constant CA,T which depends on A and T such that, for α≥α1, λ≥λ1, and for all solution of (4) the following inequality holds: (5)α3λ4∫Qe-2αηp3w2dxdadt≤C∫Qe-2αηf2dxdadt+Cα3λ4∫Qωe-2αηp3w2dxdadtwith(6)ηt,a,x=e2λΨ∞-eλΨxatT-tA-a,pt,a,x=eλΨxatT-tA-a.For the proof of this proposition see [2].Forf=0 we obtain an Observability Inequality.Lemma 3.
There existM>0 and C>0 such that the following inequality is true:(7)∫Qexp⁡-MatT-tA-aw2dxdadt≤C∫Qωw2dxdadt;herein w is the solution of system (4).Proof.
Iff=0, (5) become(8)∫Qe-2αηp3w2dxdadt≤C∫Qωe-2αηp3w2dxdadt.Choosing λ=λ1 and α=α0≥α1 such that e-2α0ηp3≤1, we obtain the result with M=2α0exp⁡(2λ1Ψ∞).In the following we consider the linear system:(9)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xbt,x+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×0,AqT,a,x=0in0,A×Ω,(10)∂φ∂t+∂φ∂a-Δφ+μφ=0inQφ=0onΣφ0,a,x=φ0in0,A×Ωφt,0,x=∫0Aβφbt,xdain0,T×Ω,where at,x and bt,x are in L∞0,T×Ω. Each system (9)-(10) is well posed and admits a unique solution. See [3] for the proof of existence and uniqueness of the solution.Remark 4.
For the linear case, that is,Gx=x can use the Hahn-Banach theorem for proving the existence of a control v which verifies q0,·,·L20,A×Ω≤ϵ, one writes(11)R0=q0,a,x;v∈L2Qωwhenqsolves9.It is easy to see that R0 is a nonempty subspace of L20,A×Ω and that (12)R⊥0is  dense  inL20,A×Ω.It suffices to show that (13)R⊥0⊂0.Suppose that (14)φ0∈R⊥0.Multiplying (9) by φ and integrating by parts, we have(15)∫QχΘh+χωvφdxdadt=∫0A∫Ωq0,a,xφ0a,xdadx.Since (16)φ0∈R⊥0,then (17)∫0A∫Ωq0,a,xφ0dadx=0.This implies that (18)∫QχΘh+χωvφdxdadt=0for every v∈L2Qω.Thenφ=0 on Θ∩ω×0,T×0,A. The unique continuation property of φ gives (19)φ=0on0,T×0,A×Ω.Therefore φ0=0. So R⊥0⊂0.However, this method cannot allow the study of the nonlinear case.


## 2.2. The Approached Controllability of the Linear System

Now, forδ>0 we consider the functional Jδ defined by (20)Jδv=12∫Qωv2dxdadt+1δ∫0A∫Ωq20,a,xdxda,where q solves(21)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xbt,x+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×ΩqT,a,x=0in0,A×Ω.Proposition 5.
The functionalJδ is continuous, strictly convex, and coercive. Therefore, Jδ reaches its minimum at a point vδ∈L2Qω. Moreover, setting qδ the associated solution of (21) and φδ the solution of (10) with φδ0,a,x=-(1/δ)qδ0,a,x has χωvδ=χωφδ and there exist C>0, K0>0 independent of δ, such that(22)∫qωφδ2dxdadt≤ChL2eM2,∫0A∫Ωqδ20,a,xdxda≤δK0hL2eM2.Remark 6.
As we will see Proposition5 solves the problem of existence of the approached sentinel for the linear case.Proof.
It is easy to check thatJδ is coercive, continuous, and strictly convex. Then it admits a unique minimizer vδ. The maximum principle gives that χωvδ=χωφδ.
Multiplying (21) with v=vδ by φδ and integrating by parts, we obtain (23)-1δ∫0A∫Ωqδ20,a,xdxda=∫QΘhφδdxdadt+∫Qωvδφδdxdadt.So (24)1δ∫0A∫Ωqδ20,a,xdxda+∫Qωφδ2dtdadx≤∫QΘhφδdxdadt.Using the inequality of Young, we obtain for any R>0(25)1δ∫0A∫Ωqδ20,a,xdxda+∫Qωφδ2dtdadx≤2RhL2eM2+12R∫Qexp⁡-MatT-tA-aφδ2dxdadt.Now, inequalities (7) and (25) give(26)1δ∫0A∫Ωqδ20,a,xdxda+∫Qωφδ2dxdadt≤2RhL2eM2+C2R∫Qωφδ2dxdadt.Choosing R such that 2R=C we have(27)1δ∫0A∫Ωqδ20,a,xdxda+12∫Qωφδ2dxdadt≤2RhL2eM2.We conclude that (28)∫Qωφδ2dtdadx≤2RhL2eM2,∫0A∫Ωqδ20,a,xdxda≤2RδhL2eM2.Proposition 7.
The functionqδ solution of system (21) verifies (29)qδL20,T×0,A;H1Ω≤ChL2eM+hL2Q.Proof.
Letλ0 be a positive constant. Put q^δ=e-λ0tqδ.
Then,q^δ solves(30)-∂q^δ∂t-∂q^δ∂a-Δq^δ+μ+λ0q^δ=βq^δt,0,xbt,x+χΘe-λ0th+χωe-λ0tφδinQq^δt,a,x=0onΣq^δt,A,x=0in0,T×Ωq^δT,a,x=0in0,A×Ω.Multiplying (30) by q^δ and integrating by parts, we obtain(31)12∫0A∫Ωq^δ20,a,xdxda+12∫0T∫Ωq^δ2t,0,xdxdt+∫Q∇q^δ2dxdadt+λ0q^δL2Q2≤12∫0T∫Ωq^δ2t,0,xdxdt+Aβ∞2b∞2q^δL2Q2+hL2Q2+Cq^δL2Q2+2RhL2eM2.Choosing λ0>Aβ∞2b∞2+C+1, we obtain (32)∇q^δL2Q+q^δL2Q≤ChL2Q+hL2eM.This gives the desired estimation.


## 2.3. Study of the Nonlinear Case

In the following we set(33)bt,x=Gη.Let Λ be an operator defined on L20,T×Ω by (34)Λη=∫0Aβaqδηda,where qδη is the solution of the following system:(35)-∂qδ∂t-∂qδ∂a-Δqδ+μqδ=βqδt,0,xGη+χΘh+χωφδinQqδt,a,x=0onΣqδt,A,x=0in0,T×0,AqδT,a,x=0in0,A×Ω.Then we have the following.Proposition 8.
The operatorΛ is continuous, bounded, and compact of L20,T×Ω. Then Λ admits a fixed point.Proof.
The proof will be done in two steps as follows.
Step  1 (boundedness and compactness of
Λ
). Let (36)Yt,a,x=∫0Aβaqδda,and denote 0,T×Ω=QT.
ThenY is the solution of the following system:(37)∂Yη∂t-ΔYη+∫0Aμβaqδda=ZηinQTYηt,x=0on0,T×∂ΩYη0,x=0inΩwith(38)Zη=∫0Aβa′qδ+χωβφδ+β2qδt,0,xGη+βhda.Under assumptions H1, H2, and H3 and the results and owing to the estimation on qδ and φδ the function Zη satisfies (39)ZηL20,T×Ω≤Cβa′∞,β∞,G∞hL2eM+hL2Q.System (37) is a heat equation where the source term and the initial condition are bounded, respectively, in L2QT and L2Ω.
ThenYη is bounded in L20,T;H01Ω and ∂Yη/∂t is bounded in L20,T;H0-1Ω. Hence, using the Lions-Aubin lemma we conclude that Λ is bounded and compact in L2QT.
Step  2 (continuity of
Λ
). Let ηn→η strongly in L2QT. Then we can show that the sequence Yηn converges to Yη strongly. For all ηn∈R, qδηn and φδηn are bounded independently to ηn. Therefore Zηn is bounded in L2QT. Then we can extract a subsequence Xηnk such that(40)Yηk⟶YηonQTZηk⟶Zηweakly  inQT∫0Aμβqδηnda⟶∫0Aμβqδηdaweakly  inL2QT.Then Yη is a solution of (37). We deduce that the sequence Yηn converges to Yη.
Since the operatorΛ is continuous, bounded, and compact on L2QT onto L2QT, Schauder’s fixed-point theorem (see [4, 5]) implies that Λ admits a fixed point.
There existsη∈L2QT such that (41)Λη=η=∫0Aβqδda.Then qδ solves(42)-∂qδ∂t-∂qδ∂a-Δqδ+μqδ=βqδt,0,xG∫0Aβqδda+χΘh+χωφδinQqδt,a,x=0onΣqδt,A,x=0in0,T×0,AqδT,a,x=0in0,A×Ω.From the foregoing, we have that φδ and qδ are bounded in L2Q.
Then we can extract a subsequence ofqδ still denoted by qδn, such that (43)qδ⇀qinL2Q,∫0Aβqδda⟶∫0AβqdainL2QT. By the same idea (44)χωφδ⇀χωvinL2Q.Since ∫0Aβqδda→∫0Aβqda, then there exists a subsequence still denoted by ∫0Aβqδda such that ∫0Aβqδda→∫0Aβqda a.e in QT.
Now sinceG is continuous, then(45)G∫0Aβqδda⟶G∫0Aβqdaa.e  inQT.Therefore, one derives that q solves the following system:(46)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xG∫0Aβqda+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×0,AqT,a,x=0in0,A×Ω,and we have also, for δ→0, q0,a,x=0 a.e. in 0,A×Ω.


## 3. Application to the Detection of the Incomplete Parameter for a Nonlinear Population Dynamics Model

For a given positive real functionF, we consider the following nonlinear population dynamics model:(47)∂y∂t+∂y∂a-Δy+μy=λξ^inQyt,a,x=0onΣyt,0,x=F∫0Aβayλ,τdain0,T×Ωy0,a,x=y0a,x+τy^0a,xin0,A×Ω,where Ω is a bounded open subset of RN, N≥1, with a smooth boundary ∂Ω. Let T be a positive real and let ω be an open subset such that ω¯⊂Ω. Here, yt,a,x is the distribution of individuals of age a at time t and location x∈Ω, χω is the characteristic function of ω, A is the maximal live expectancy, Δ is the Laplacian with respect to the spatial variable, and βa and μa denote, respectively, the natural fertility and the natural death rate of individuals of age a. Thus, the formula ∫0Aβayt,a,xda denotes the distribution of newborn individuals at time t and location x. In an oviparus species it denotes the total eggs at time t and position x. Therefore, the quantity F∫0Aβayt,a,xda is the distribution of eggs that hatch at time t and position x.System (47) describes the evolution of internal controlled age and space structured population under inhospitable boundary conditions in the case that the flux of individuals has the form -∇yt,a,x.In system (47) y0 is given in L20,A×Ω.The data of system (47) are incomplete in the following sense:(i)
y^0∈L20,A×Ω is unknown and y^0L20,A×Ω=1;(ii)
ξ^∈L2Q is unknown and ξ^L2Q=1;(iii)
λ∈R is unknown and small enough;(iv)
τ∈R is unknown and small enough.We denote by yobs=m0 the observation that is a function in L2QΘ, where Θ⊂Ω is the observation set. We know that (47) has a unique solution that we denote by yλ,τ=yt,s,x,λ,τ in some relevant space. The question is as follows:How to calculate the incomplete termλξ^ independently from the variation τy^0 around the initial data?The question above is natural and leads to some developments. An answer is given by the least squares method. This method consists in considering the unknowns λξ,τy0=ν,ϑ, as control variables; then the state yt,s,x,ν,ϑ has to be driven as close as possible to m0. So one studies an optimal control problem. By this way one looks to the pair ν,ϑ but unfortunately there is possibility to find ν or ϑ independently. The sentinel method of Lions [6] is a particular least squares method which is adapted to the identification of parameters in ecosystems with incomplete data. Many papers are devoted to this topic in the literature. The sentinel concept relies on the following three objects: some state equation (e.g., (47)), some observation function yobs, and some control function v to be determined. Many papers use the definition of Lions in the theoretical aspect (see, e.g., Bodart [7], Bodart and Fabre [8, 9]), as well as in the numerical one (see Bodart and Demeestere [10], Demeestere [11], and Kernevez [12]).In [13] the author studies the detection of incomplete parameter for a linear population dynamic model. In this paper we are concerned by the nonlinear case. Moreover our method allows us to easily address the numerical aspect.We suppose thatF∈C1R and F′∈L∞R.Proposition 9.
The functionsλ↦yλ,τ and τ↦yλ,τ are differentiable at the point 0.Proof.
Let(48)y^t,a,x=e-λ0tyλ,τ-y0,where yλ,0=y0.
It is easy to prove thaty^ is the solution of the following system:(49)∂y^∂t+∂y^∂a-Δy^+μ+λ0y^=0inQy^t,a,x=0onΣy^t,0,x=e-λ0tF∫0Aβyλ,τda-F∫0Aβy0day^0,a,x=τy^0a,x.Multiplying (49) by y^ and integrating by parts, we obtain (50)12∫0A∫Ωy^2T,a,xdxdt+12∫0T∫Ωy^2t,A,xdxda+∫Q∇y^2dxdadt+∫Qμ+λ0y^2dxdadt(51)=12∫0A∫Ωτ2y^02dxda+12∫0T∫Ωy^2t,0,xdxdt.We have(52)y^t,0,x=e-λ0tF∫0Aβyλ,τda-F∫0Aβy0da.This implies (53)y^t,0,x2≤K∫0Aβy^2;then(54)∫0T∫Ωy^2t,0,xdxdt≤Kβ∞2y^L2Q2.Choosing λ0=1+Kβ∞2 and using equality (51) we obtain (55)∇y^L2Q2+y^L2Q2≤Cτ2.This gives (56)yλ,τ⟶y0a.e  onQuniformlyand then (57)∫0Aβyλ,τda⟶∫0Aβy0daa.e  onQTuniformly.Now we consider(58)∂z∂t+∂z∂a-Δz+μ+λ0z=0inQzt,a,σ=0onΣzt,0,x=F′∫0Aβy0da∫0Aβe-λ0tzdain0,T×Ωz0,a,x=y^0a,xin0,A×Ω;zτ=y^/τ and pτ=zτ-z solving(59)∂pτ∂t+∂pτ∂a-Δpτ+μ+λ0pτ=0inQpτt,a,σ=0onΣpτt,0,x=1τe-λ0tF∫0Aβyλ,τda-F∫0Aβy0da-F′∫0Aβy0da∫0Ae-λ0tβzdain0,T×Ωpτ0,a,x=y^0a,xin0,A×Ω.From the regularity of F and the uniform convergence of τ↦yλ,τ, we have (60)∀ϵ>0,∃τ0such that∀τ<τ0;e-λ0tτF∫0Aβyλ,τda-e-λ0tτF∫0Aβy0da-F′∫0Aβy0da∫0Aβe-λ0tzda≤C∫0Aeλ0tβpτda+ϵ∫0Ae-λ0tβzτda.Multiplying (59) by pτ and integrating by parts, we obtain(61)12∫0A∫Ωpτ2T,a,xdxdt+12∫0T∫Ωpτ2t,A,xdxda+∫Q∇pτ2dxdadt+∫Qμ+λ0pτ2dxdadt=12∫0T∫Ωpτ2t,0,xdxdt.Using (60), we obtain(62)12∫0A∫Ωpτ2T,a,xdxdt+12∫0T∫Ωpτ2t,A,xdxda+∫Q∇pτ2dxdadt+∫Qμ+λ0pτ2dxdadt≤Cβ∞2ϵ+Aβ∞2pτL2Q2.Choosing λ0 such that λ0>1+Aβ∞2 we obtain (63)pτL20,T×0,A;H1Ω2≤Cϵ.This implies pτ→0 a.e. in L20,T×0,A;H01Ω as τ→0. Then zτ→z a.e. in L20,T×0,A;H01Ω.
Thereforeτ↦e-λ0tyτ,λ is differentiable at 0. We conclude that τ↦yλ,τ is differentiable at 0.
The proof of the differentiability of the functionλ↦yλ,τ goes similarly.Now leth be some function in L2QΘ and let Qω be an open and nonempty subset of Q. For a control function v∈L2Qω, we define the functional(64)sλ,τ=∫QΘhyλ,τdxdadt+∫Qωvyλ,τdxdadt.Using Proposition 2, one has(65)∂S∂τ0,0=∫QΘhyτdxdadt+∫Qωvyτdxdadt,where(66)yτ=limτ→0⁡y0,τ-y0,0τ is the solution of the system(67)∂yτ∂t+∂yτ∂a-Δyτ+μyτ=0inQyτt,a,x=0onΣyτt,0,x=∫0AβF′∫0Aβy0,0dayτdain0,T×Ωyτ0,a,x=y^0a,xin0,A×Ω.We say that S defines a sentinel for problem (47) if there exists v such that S is insensitive (at first order) with respect to the missing terms τy^0; this means(68)∂S∂τ0,0=0for any y^0 and if the norm vL2Qω of v is minimal.The functionalS assumed to be nonzero is an approached sentinel if, for any δ>0, there exists v∈L2Qω such that(69)∂S∂τ0,0=∫QΘhyτ0,0dxdadt+∫Qωvyτ0,0dxdadt≤δ2.The existence of sentinel is equivalent to a controllability problem.Proposition 10.
Let us consider the following system:(70)-∂q∂t-∂q∂a-Δq+μq=βqt,0,xF′∫0Aβy0,0da+χΘh+χωvinQqt,a,x=0onΣqt,A,x=0in0,T×0,AqT,a,x=0in0,A×Ω,where y0 is the solution of system (47) with λ=τ=0. The approached sentinel condition (69) is equivalent to find v∈L2Qω such that (71)q0,·,·L20,A×Ω≤δfor any δ>0. The exact sentinel condition is equivalent to finding v∈L2Qω such that (72)q0,a,x=0a.ein0,A×Ω.Proof.
Multiplying (70) by yτ and integrating by parts, we obtain(73)∫0A∫Ωq0,a,xy^0dxdadt=∫QΘhyτdxdadt+∫Qωvyτdxdadt.And then the approached sentinel condition (74)∫QχΘh+χωvyτdtdadx≤δ2is equivalent to (75)∫0A∫Ωq0,a,xy^0dxda≤δ2for every(76)y^0∈L20,A×Ω.That is equivalent to (77)q0,a,xL20,A×Ω≤δ.We have the following result.Proposition 11.
Leth∈L2eM, if F∈C1R and F′∈L∞R; then under assumptions H1, H2, and H3 there exists a unique control v∈L2Qω satisfying (68).Proof.
Replacingbt,x in (21) by F′∫0Aβy(0,0)da we obtain the approached sentinel, and we obtain the sentinel when δ goes to zero.


## 4. Information Provided by the Sentinel

The function(78)yλ=limλ→0⁡yλ,0-y0,0λis the solution of the system(79)∂yλ∂t+∂yλ∂a-Δyλ+μyλ=ξ^t,a,xin0,T×0,A×Ωyλt,a,x=0onΣyλt,0,x=∫0AβF′∫0Aβy0,0dayλdain0,T×Ωyλ0,a,x=0in0,A×Ω.We have (80)Sobsλ,τ≃S0,0+λ∂S∂λ0,0+τ∂S∂τ0,0⟺Sobsλ,τ-S0,0≃λ∂S∂λ0,0.Multiplying system (22) by yλ and integrating by parts, we obtain (81)∫Qqξ^dxdadt=∫QχΘh+χωvyλdxdadtand then(82)∫Qλqξ^dxdadt=∫QχΘh+χωvm0-y0,0dxdadt.Knowing h, v, m0, and y(0,0), q can be calculated. Thus we obtain an integral equation in λξ^.



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*Source: 2820613-2016-03-03.xml*