Null controllability and application to data assimilation problem for a linear model of population dynamics

In this paper we study a linear population dynamics model. In this model, the birth process is described by a nonlocal term and the initial distribution is unknown. The aim of this paper is to use a controllability result of the adjoint system for the computation of the density of individuals at some time T . Contrôlabilité à zéro et application à un problème d’assimilation de données sur un modèle linéaire de dynamique des populations. Résumé Dans cet article nous étudions un modèle linéaire de dynamique des populations. Dans ce modèle, le processus de naissance est défini par un terme non local et la distribution initiale des individus n’est pas connue. L’objectif ici est d’utiliser un resultat de contôlabilité du système adjoint pour la détermination de la densité des individus à un instant T .


Introduction
We consider a population living in a bounded open set Ω of R N , N ≥ 1.The boundary of Ω that is Γ, is assumed to be sufficiently smooth.Let y(t, a, x) be the distribution of individuals of age a at time t and location x ∈ Ω and let T be a positive constant.In the sequel µ 0 (t, a, x) and β(t, a, x) stand respectively for the natural death and birth rate of individuals of age a at time t and location x.We assume that the boundary Γ is inhospitable.If the flux of individuals reads −∇y where ∇ is the gradient with respect to the spatial variable, then y solves the following system ∂y ∂t + ∂y ∂a − y + µ 0 y = 0 in (0, T ) × (0, A) × Ω y(t, a, x) = 0 on (0, T ) × (0, A) × Γ y(0, a, x) = y 0 (a, x) in (0, A) × Ω y(t, 0, x) where ∆ is the laplacian with respect to the spatial variable and y 0 (a, x) is the initial distribution of individuals of age a at location x.When this initial distribution y 0 , is known one can use an integration along charateristic lines and an orthonormal basis of eigenfunctions of the laplacian to compute y(T, a, x), see [4].In this paper we suppose that the initial distribution y 0 is unknown and we want to compute y(T, ., .)using some observations on the state y during an interval of time (0, T ).This is in fact a data assimilation problem.More precisely, the problem is to predict the density of individuals at some t > T from the knowledge of some observations during an interval of time (0, T ).
The classical way to solve such problem is to compute first the initial distribution.This kind of problem is generally ill posed and requires traditionally, Tychonof regularization and minimization of a quadratic functional.[9], [7].
As soon as the initial distribution is determined, one can compute y(t, ., .) in a classical way.
Here, we want to compute first y(T, ., .)and afterwards, one can use it as a "'new initial distribution"' for the study of y(t, ., .), for t > T .
The problem of recovering unknown data in population dynamics model was extensively studied.In [12], the author performed a technique for recovering the natural death rate in a Mc Kendrick model.The method there uses an overdeterminated data y(T, a) = ψ(a) and the explicit form of the solution.In [5] the problem is also to recover the natural birth and death rates from the knowledge of the initial and final distribution.In [8], the goal is different from the previous one.More precisely in [8] the authors studied a method for determining the individual survival and reproduction function from data on population size and cumulative number of birth in a linear population model of Mc Kendrick type.These goals are quite different from the one we study here.Our method uses essentially a null controllability result of an adjoint problem.Similar problem in the framework of parabolic equation was studied earlier by JP Puel in [11].
In [14], an application of the approximate controllability property to data assimilation problems was studied by the author.The question adressed there is whether one can use an approximate controllability result for recovering the initial data for a linear population dynamics model.A lot of papers are devoted to the study of null controllability property for population dynamics models.In [3], a null controllability result was established for a linear population dynamics.The method used a fixed point theorem and Carleman inequality for parabolic equations.Here, we will establish a new observability inequality with a weight.This result allows us to control on the whole domain (0, A) × Ω .The remainder of this paper is as follows: in Section 2, we state assumptions and prove the null controllability result.The Section 3 is devoted to the statement of an approximation method for computing the distribution at time T .
We have also included an Appendix, where we give the proof of the Carleman inequality with the careful study of the dependence of the constants on s, λ, T and A.

Assumptions and null controllability results
We state first the hypotheses which will be used.
The following notations will be used in the sequel: The main goal of this section is to prove a null controllability result.First, let us consider the system where where π(a) = exp(− a 0 µ 1 (s) ds) and β = πβ.The problem is reduced to find for any such that the corresponding solution p verifies (2.2).On the other hand it is obvious that β verifies A 2 .In this section, we will consider the previous system and we write β instead of β.
We want now to give a Carleman inequality from which we will derive an observability inequality.

Proposition 2.4.
There exist positive constants λ 0 > 1 and C(Ψ) such that for λ > λ 0 and such that for all solution z of (2.4) the following inequality holds: Remark 2.5.The proof of this Carleman inequality follows the method of [10] for parabolic equation.In [13] we have established similar Carleman inequality, but without the particular form of the constants.See also [2].
For completeness and in order to justify the particular form of the constants λ 0 and s 0 (λ) we give the entire proof in the appendix, at the end of the paper.
The goal now, is to derive from the Carleman inequality the following observability inequality which is helpful for the proof of Theorem 2.2.Proposition 2.6.Suppose that A 1 − A 2 are fulfilled.Then there exists a positive constant C depending only on a 0 , A, Ψ, Ω and T such that where and s > s 0 (λ).
Proof.Let us assume first that T > A. We will prove in this case that Then, since z solves (2.4) it follows that q solves the system: Multiplying (2.9) by q and integrating the result over (0, T − σ) × Ω, we get: (2.10) Since q(0, x) = z(σ, 0, x), using (2.10) and (2.4), and thanks to the Cauchy Schwarz inequality we get: where This means : An integration on (T − A, T ) with respect to σ yields: Therefore:

13) we get:
A ( Using now (2.5), and setting Note that (2.15) holds for all t ≤ T such that t − A > .Let now σ ∈ (T − , T ).On the one hand we have σ − A > T − A − > .On the other hand, let us consider system (2.9).Inequality (2.11) yields: (2.17) Combining (2.15) and (2.17) we obtain: Integrating now both sides of (2.18) over (T − , T ) with respect to the variable σ, one deduces after a standard change of variables: ).We suppose now that T < A. Let σ ∈ (0, A − T ).Consider the characteristic line C = {(t, σ + t); t ∈ (0, T )} and set q(t, x) = z(t, σ + t, x).It follows immediately that q solves the following system: Using now the standard observability inequality for the heat equation [6], we infer that: This is equivalent to: Integrating both sides over (0, A − T ) with respect to the variable σ gives: Making the following change of variables: a = σ + T in the left hand term, and a = σ + t in the right hand term we get: (2.24) Let us now take σ ∈ (0, T ) and consider the following characteristic line C = {σ + a, a), a ∈ (0, T − σ)}.Let q(a, x) = z(σ + a, a, x).One can see that q solves the system (2.25) below: Let us multlply (2.25) by q and integrating over (0, T − σ) × Ω.We obtain recalling system (2.4): From the assumption A 2 and the boundedness of η and ϕ on (a 0 , a 1 ) × Ω we have: where Then, we deduce after an integration over (0, T ) with respect to the variable σ: Using now the last definition of q;inequality (2.5) and setting a = T −σ, we get: The proof is now complete.
Let us prove now the Theorem 2.2.
Proof.We assume that A 1 −A 2 are satisfied.For g ∈ L 2 (Q A ), we introduce for α > 0 the functional J α defined on L 2 (Q ω ) by: where p solves (2.3).The functional J α is continuous, convex and verifies: Consequently, J α achieves its minimum at a unique point v α .Moreover the maximum principle gives: where z α solves (2.30) and p α is the solution to Multiplying (2.31) by z α and integrating over Q we obtain: Then, This yields using inequality (2.6) and (2.29) (2.32) Consequently, the sequence Then, when α → 0, after extraction of a subsequence, the sequence still denoted (v α , p α ) converges weakly towards (v, p), which solves (2.3).Particularly, we have p α (0, ., .)p(0, ., .) in L 2 ((0, A) × Ω) weakly so that p verifies (2.2).Note that the following inequality holds too: (2.33) Let us prove that (v α , p α ) converges strongly to Since v α is the unique minimizer of J α , we infer that Consequently, the weak convergence of v α towards v, yields that v α converges strongly to v in L 2 (Q ω ) as n goes to ∞.This implies obviously that p α converges strongly to p in L 2 (Q).So, the sequence This ends the proof.

Recovery of the state value y(T )
We give here our data assimilation result.This result uses mainly the null controllability result proved above.Next, we give a possible approximation method of the null controllability problem by means of some optimal control problems.Beforehand, we will first prove the following proposition.
Proof.We will prove the proposition when A > T , the case A < T can be proved using analogous arguments.This proof will be done in two steps: Step 1: construction of an adapted countable and dense set.
It follows that for all m, O m is a bounded subset of O. Now, let P be the set of all polynomials on R N +1 with rational coefficients.Let P m = {f 1 Om ; f ∈ P }.The set G = ∪ m≥1 P m is countable and is dense in L 2 (O).
See [1, page 29].Let us consider f ∈ G, there exists an integer m such that f ∈ P m .Let g = ρ −1 (a)f .From the defintion of P m it follows now that the function g ∈ L 2 (O).Furthermore, writting Step 2: construction of the orthonormal basis.
Let us write f 1 , f 2 , ... the functions of F .It suffices to extract from this sequence an infinite and dense sequence of linearly independent elements, and after to apply the orthogonalization method.For this aim, we exclude from the sequence (f k ) all function f j which can be represented as a linear combinaison of f i with i < j.We obtain thus doing, the desired sequence.The proof is complete.
We assume that the initial distribution y 0 belongs to L 2 (Q A ) .This assumption is natural since y stands for the density of the population.Therefore, it follows that y(T, ., .)∈ L 2 (Q A ). Now, let us consider an orthonormal basis of the form (ρg k ) with g k ∈ L 2 (Q A ).Then, on the one hand, we have: where y k = A 0 Ω y(T, a, x)ρ(a)g k (a, x) dx da.On the other hand, for all k, by virtue of Theorem 2.2, there exists ṽ(g k ) ∈ L 2 (Q ω ) such that the associated solution p of (2.1) verifies (2.2).Then, multiplying (1.1) by p and integrating the result over Q, we obtain Therefore, This equation gives the coefficients of the desired state value y(T ) from the measurements of the solution on the subset ω.At the same time, if we use an approximation of the exact value of y on ω, this formula describes the effect of the error on the coefficients of y(T ).
Using (3.1), we get thanks to the Cauchy Schwarz inequality: . This yields Note that (3.3) is a stability inequality.We now, summarize the method for retrieving the state value y(T ) in the following Proposition.

Proposition 3.2. Let us consider an orthonormal basis of L
O. Traore and iii) The state at time T , y(T, ..) is given by where

Concluding Remark
This paper adresses the essential problem of data assimilation.Here, we have shown that from the knowledge of the density of individuals on a small open set and during the interval of time (0, T ), one can compute y c := y(T, a, x), the density at the time T .From this, we can now compute this density at any time t such that T < t < T by means of the following system: The method we have been studying here gives a theoretical result but it could also be used for a pratical recovery of the state value y(T ) from measurements of the solution on a small open set.We then have to recover an approximation of y(T ) on a finite dimensional basis.The choice of this basis is crucial as it has to provide a good approximation for y(T ) but it has to contain a small number of elements to minimize the adjoint control problems to be solved.This will be the subject of a forthcoming work and we will compare the results given by this method with classical methods using Tychonov regularization.

Appendix: proof of Proposition 2.4
Here, we suppose that the function z ∈ C 2 (Q) and verifies where f ∈ L 2 (Q) and we prove the following more general Carleman inequality Proposition 5.1.There exist positive constants λ 0 > 1 and C(Ψ) such that for λ > λ 0 and and all solution z of (5.1) the following inequality hold: Taking f = 0 one obtains inequality (2.5).
We have by integration by parts: Hence using (5.5)and ( 5.3) it follows: (5.16) An integration by parts leads to: (5.17) This gives Keeping in mind (5.3), an integration by parts with respect to the variable t yields: Likewise, one gets easily that: and (5.21) Now, we are concerned by the term I 3,j .
Furthermore, we have: Then, it follows from (5.15) and (5.34): (5.36) Recalling (5.30)-(5.31),one can choose s and λ sufficiently large so that This means more precisely that there exists positive constants λ 0 > 1 such that: Furthermore since (5.37) Actually, we want now to eliminate in (5.37) the term 2sλ 2 δ 2 q ϕ |∇u| 2 dQ.For this aim, we introduce a cut-off function α such that: α ∈ C ∞ 0 (ω); 0 ≤ α ≤ 1; and α = 1 on ω.Multiplying P 2 u by ϕα 2 u and integrating the result over Q leads to: where C is a positive constant.Now, since ϕ ≤ Cϕ 3 with C a positive constant, and using the properties of α and Ψ we deduce: (5.42) Therefore we deduce from the previous estimate that:
Remark 2.1.Assumptions A 1 and A 2 are classic in the study of population dynamics.Indeed, A 0 µ 1 (a) da = +∞ means that the survival likelihood of individuals, that is exp(− a 0 µ 1 (s) ds) tends towards zero as a goes to A. In other words, all individuals die before the age A. Assumption A 2 means that the young and the old individuals are not fertile.
and C(A, β, . ..) are several positive contants depending on A, β, . . .Sometimes we will write dQ instead of dt da dx.