Laurent Bourgeois
email: laurent.bourgeois@ensta.fr

QUANTIFICATION OF THE UNIQUE CONTINUATION PROPERTY FOR THE HEAT EQUATION

Keywords: 2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35 Stability estimate, Lateral Cauchy data, Carleman estimate, Distance function

In this paper we prove a logarithmic stability estimate in the whole domain for the solution to the heat equation with a source term and lateral Cauchy data. We also prove its optimality up to the exponent of the logarithm and show an application to the identification of the initial condition and to the convergence rate of the quasi-reversibility method.

1.

Introduction. This paper deals with the ill-posed problem of finding the solution to the heat equation from a source term and lateral Cauchy data. To be precise, let us denote by Ω a smooth domain of R d , d ≥ 1, Γ 0 an open subset of ∂Ω and n the outward unit normal to Ω. Given T > 0, some function f in Ω × (0, T ) and two functions (g 0 , g 1 ) on Γ 0 × (0, T ), our problem consists in finding u in Ω × (0, T ) such that    ∂ t u -∆u = f in Ω × (0, T ) u = g 0 on Γ 0 × (0, T ) ∂ n u = g 1 on Γ 0 × (0, T ).

(

) 1
Problems such as [START_REF] Bourgeois | The "exterior approach" applied to the inverse obstacle problem for the heat equation[END_REF] arise in the framework of inverse problems related to the heat equation, where f = 0 and the data (g 0 , g 1 ) correspond to measurements on the accessible part Γ 0 of the boundary of Ω (see for example [START_REF] Bourgeois | The "exterior approach" applied to the inverse obstacle problem for the heat equation[END_REF][START_REF] Bécache | Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case[END_REF]). The uniqueness property for that problem, that is (f, g 0 , g 1 ) = 0 ⇒ u = 0, is well-known and is a consequence of Holmgren's Theorem (see for example Theorem 5.3.3 in [START_REF] Hörmander | Linear Partial Differential Operators[END_REF]). However it is also well-known that this problem is ill-posed in the sense of Hadamard, due to the fact that the existence property does not hold, even for smooth data. More difficult is the question of stability, that is the quantification of the uniqueness property. This question for our problem is the following: if f , g 0 and g 1 are small (instead of being zero), let say smaller than δ, how small is u with respect to δ ? This question has been studied for a long time and is part of the monographs or review papers [START_REF] Lavrentiev | Ill-posed problems of mathematical physics and analysis[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[END_REF][START_REF] Vessella | Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates[END_REF][START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF], which all refer to a long series of contributions. In those contributions the authors obtain stability estimates which may have different forms and depend on the regularity assumptions on the domain Ω and on the functions f , g 0 and g 1 . In particular, it is well-established that the solution u has a Hölder dependence on the data (f, g 0 , g 1 ) in any subdomain Ω ε × (ε, T -ε) of Ω × (0, T ), where Ω ε is the subset of Ω excluding the points such that their distance

LAURENT BOURGEOIS

to the complementary part of Γ 0 in ∂Ω are smaller than ε > 0. In this vein, we refer for example to [START_REF] Vessella | Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates[END_REF] (see Theorem 3.5.1) or to [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] (see Theorem 5.1), where the stability estimates are obtained by using a Carleman estimate applied to solutions to the heat equation, even if these two papers propose different ways to exploit such Carleman estimate. It is worth noting that in the result of [START_REF] Vessella | Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates[END_REF], the time interval is not truncated near the final time T . These Hölder-type stability estimates imply the uniqueness property but are stronger since they quantify the unique continuation.

In [START_REF] Klibanov | Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data[END_REF], a logarithmic stability estimate is obtained in the whole domain Ω × (0, T ) but only in the case when the support of the lateral Cauchy data Γ 0 × (0, T ) is the whole lateral boundary ∂Ω × (0, T ). In other words, the objective of [START_REF] Klibanov | Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data[END_REF] amounts to recover the initial condition, since all the spatial boundary data are known. In [START_REF] Boulakia | Quantification of the unique continuation property for the nonstationary Stokes problem[END_REF], a logarithmic estimate is obtained in a subdomain Ω × (ε, T -ε) when Γ 0 ∂Ω for the non-stationary Stokes system instead of the heat equation, which is a similar but more complicated case. Thus, the same kind of estimate in Ω × (ε, T -ε) could be derived in the case of the heat equation by using the arguments of [START_REF] Boulakia | Quantification of the unique continuation property for the nonstationary Stokes problem[END_REF]. However, to the author's knowledge, a stability estimate for the heat equation in the whole domain Ω × (0, T ) with lateral Cauchy data supported in Γ 0 × (0, T ) with Γ 0 ∂Ω is unknown.

In this paper we obtain such a stability estimate in the whole domain Ω × (0, T ) in the particular case when Ω and Γ 0 satisfy the following assumption. Let us denote Q = Ω × (0, T ), Σ 0 = Γ 0 × (0, T ) and P = ∂ t -∆. We obtain the following logarithmic stability estimate.

Main Theorem. For all s ∈ (0, 1), there exists a constant C > 0 such that for all u ∈ H 1 (0, T ;

H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), ||u|| L 2 (0,T ;H 1 (Ω)) ≤ C ||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) log s 2 + ||u|| H 1 (0,T ;H 1 (Ω)) +||u|| L 2 (0,T ;H 2 (Ω)) ||P u|| L 2 (Q) +||u|| H 1 (Σ 0 ) +||∂nu|| L 2 (Σ 0 ) .
As exposed in Remark 5, such theorem may be considered as an extension of the logarithmic stability estimates obtained for the Laplace equation in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de Laplace[END_REF] (for a domain of class C ∞ ) and [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 , M2AN Math[END_REF] (for a domain of class C 1,1 ). Other logarithmic estimates for the conductivity equation or/and non-smooth domains may be found in [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF][START_REF] Takeuchi | Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation[END_REF][START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains[END_REF]. The proof of the Main Theorem consists of two steps. The first one is a stability estimate in the subdomain Ω ε × (ε, T -ε) obtained with the help of a Carleman estimate. The second step consists in completing the previous estimate in the whole domain Ω × (0, T ) with the help of a Hardy-type inequality. Such general scheme was first introduced in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de Laplace[END_REF] for the Laplace equation. The main issue in step 1 consists in exhibiting in the estimate the explicit dependence of the constants with respect to ε. To this aim we need to know the explicit dependence of the constants in the Carleman estimate with respect to the length of the time interval, as it is done in [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF][START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF]. To obtain the Carleman estimate, we proceed similarly as in the pioneering contribution [START_REF] Fursikov | Controllability of evolution equations[END_REF], that is by using some Carleman weight that vanishes when t(T -t) → 0. Such kind of weight is also used in [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF][START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF] and differs from the one used in [START_REF] Lavrentiev | Ill-posed problems of mathematical physics and analysis[END_REF][START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF]. As in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF][START_REF] Boulakia | Quantification of the unique continuation property for the nonstationary Stokes problem[END_REF], we apply the Carleman estimate in many time intervals covering (0, T ) up to the small parameter ε and the length of which tends to 0 when ε → 0, but the novelty here consists in deriving an estimate in each interval, the constants of which depend explicitly on ε. Another novelty of this paper consists in constructing a smooth Carleman weight which close to boundary ∂D or ∂O coincides with the distance function to such boundary. The combination of such idea and the geometric assumption 1.1 considerably simplifies the proof. The possibility of getting rid of Assumption 1.1 is discussed at the end of the paper.

Our article is organized as follows. The second section is devoted to the construction of a useful Carleman weight with the help of a slight modification of Theorem 9.4.3 in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]. The third one is devoted to the derivation of the Carleman estimate while the Main Theorem is proved in section 4 and its optimality discussed in section 5. We then show that the Main Theorem enables us to estimate the initial condition (section 6) and to obtain a convergence rate for the method of quasi-reversibility (section 7). We complete the paper by some remarks about the geometric assumption 1.1.

2. Construction of a Carleman weight. In order to derive a useful Carleman estimate, we will need the following theorem. Theorem 2.1 is a slight refinement of Theorem 9.4.3 in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]: the only difference is that the second property in Theorem 2.1 is replaced by the weaker property η(x) = 0 for all x ∈ ∂D. The proof of Theorem 9.4.3 in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] contains two steps. The first one is a construction of a smooth positive function v which vanishes on the boundary of D and with no critical points in a tubular vicinity of the boundary of D. The second step consists in moving the critical points of v into the domain O with the help of a well chosen map which preserves the other properties of the function v. The proof of Theorem 2.1 follows the same lines. The only difference with the proof of Theorem 9.4.3 in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] (see Chapter 14) is the replacement of the so-called Step 1 in the proof of the intermediate Theorem 14.2.3 in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] by the following lemma. Proof. Let us denote, for x ∈ D, d(x) = d(x, ∂D) the distance function to the boundary ∂D. Such function is not smooth in D but from [START_REF] Gilbarg | Elliptic Partial Differential Equations od Second Order[END_REF] (see Lemma 14.16), there exists an open neighborhood N 0 of ∂D in R d such that d ∈ C m (V 0 ) where V 0 = N 0 ∩ D and ∇d(x) = -n(x ) for x ∈ V 0 , where x is the unique point in ∂D such that d(x) = |x -x | and n is the outward unit normal vector. In particular

|∇d(x)| > 0 for x ∈ V 0 . Now let us consider two smaller open neighborhoods N N 1 N 0 of ∂D in R d and V = N ∩ D, V 1 = N 1 ∩ D. Let us introduce the cutoff function φ ∈ C ∞ 0 (D) with 0 ≤ φ ≤ 1, φ = 1 on the closure of D \ V 1 and φ = 0 on V. The function v defined in D by v = φ + (1 -φ)d
satisfies all the requirements of the lemma.

Classically, the function η derived in Theorem 2.1 for m = 2 is used to build an adapted weight for a Carleman inequality that will be then used to obtain some controllability results (see for example [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]). ) -e 4λL θ(t) , ξ(x, t) = e λ(2L+η(x)) θ(t) ,

with L = ||η|| ∞ and θ(t) = t(T -t). We denote Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ). The aim of this section is to prove the following Carleman estimate in Q.

Theorem 3.1. There exist some constants λ 0 , ρ 0 > 0 and C > 0 such that for all T , for all λ > λ 0 , ρ > ρ 0 and all u ∈ C 2 (Q),

s 3 λ 4 Q ξ 3 u 2 e 2sφ dxdt + sλ 2 Q ξ|∇u| 2 e 2sφ dxdt ≤ C Q (P u) 2 e 2sφ dxdt +C s 3 λ 3 Σ ξ 3 u 2 e 2sφ dsdt + C sλ Σ ξ|∇u| 2 e 2sφ dsdt + C 1 sλ Σ 1 ξ (∂ t u) 2 e 2sφ dsdt,
where s = ρ(T + T 2 ).

Remark 1. It is important to note that in the Carleman estimate above the constants λ 0 , ρ 0 and C are independent of T , this is why we call it a uniform Carleman estimate.

Proof of Theorem 3.1. We here reproduce the proof of Lemma 1.3 in [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF] by using the same notations. However there are two differences: contrary to that lemma, on the one hand we do not assume that u = 0 on the boundary Σ and on the second hand we have ∇η = 0 in the whole domain Ω instead of in a subdomain. The consequence is that the right-hand side of the Carleman estimate involves some boundary terms instead of a volume term in a subdomain. We will only insist on the boundary terms and refer to [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF] for the remainder of the proof. Let us denote w = e sφ u, that is u = e -sφ w. We compute As a consequence, we have

∂ t u = (∂ t w -s ∂ t φ w)e -
(M 1 w, M 2 w) L 2 (Q) ≤ ||e sφ P u|| 2 L 2 (Q) + ||M 3 w|| 2 L 2 (Q) .
We observe that M 1 w and M 2 w are both sums of three terms, so we write with obvious notations

(M 1 w, M 2 w) L 2 (Q) = 3 i,j=1 I ij , with I 11 = -2s 3 λ 4 Q ξ 3 |∇η| 4 w 2 dxdt, I 21 = -2s 3 λ 3 Q ξ 3 |∇η| 2 (∇η • ∇w)w dxdt, I 31 = -s 2 λ 2 Q ξ 2 |∇η| 2 w ∂ t w dxdt, I 12 = -2sλ 2 Q ξ|∇η| 2 w ∆w dxdt, I 22 = -2sλ Q ξ(∇η • ∇w)∆w dxdt, I 32 = - Q ∂ t w ∆w dxdt, I 13 = -2s 2 λ 2 Q ξ|∇η| 2 ∂ t φ w 2 dxdt, I 23 = -2s 2 λ Q ξ∂ t φ(∇η • ∇w)w dxdt, I 33 = -s Q ∂ t φ w ∂ t w dxdt.
After many applications of the Green Theorem and the fact that w and all its derivatives vanish at t = 0 and t = T since φ → -∞ for x ∈ Ω when t → 0 + or t → T -, we obtain

(M 1 w, M 2 w) L 2 (Q) = Q V (w) dxdt - Σ B(w) dsdt
by separating volume and boundary terms. While the terms I 11 , I 31 , I 13 , I 33 do not produce any boundary term (see equations (1.31), (1.34), (1.48) and (1.52) in [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]), the terms I 21 , I 12 , I 22 , I 32 and I 23 produce the opposites of the following boundary terms (see equations (1.32), (1.37), (1.41), (1.45) and (1.49) in [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]):

B 21 = s 3 λ 3 Σ ξ 3 |∇η| 2 ∂ n η w 2 dsdt, B 12 = 2sλ 2 Σ ξ|∇η| 2 ∂ n w w dsdt, B 22 = 2sλ Σ ξ(∇η • ∇w)∂ n w dsdt, B 32 = Σ ∂ t w ∂ n w dsdt, B 23 = s 2 λ Σ ξ∂ t φ ∂ n η w 2 dsdt, so that Σ B(w) dsdt = B 21 + B 12 + B 22 + B 32 + B 23 .
Concerning the volume terms, it is proved in [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF], more precisely in the Step 2 of the proof of Lemma 1.3, that there exist some constants λ 0 , ρ 0 > 0 and c > 0 such that for all T , for all λ > λ 0 , ρ > ρ 0 and all

u ∈ C 2 (Q), Q V (w) dxdt -||M 3 w|| 2 L 2 (Q) ≥ c s 3 λ 4 Q ξ 3 w 2 dxdt + sλ 2 Q ξ|∇w| 2 dxdt .
In particular, this estimate uses the fact that |∇η| ≥ c in Ω for some constant c > 0.

It remains to estimate the boundary terms.

|B 21 | ≤ C s 3 λ 3 Σ ξ 3 w 2 dsdt, |B 22 | ≤ C sλ Σ ξ|∇w| 2 dsdt.
By applying a Young's inequality to B 12 , we obtain

|B 12 | ≤ s 2 λ 3 Σ ξ 2 w 2 dsdt + λ Σ |∇w| 2 dsdt.
From the fact that ξ ≥ 4/T 2 and s ≥ ρ T 2 , we obtain that sξ ≥ 4ρ, so that for large ρ,

|B 12 | ≤ C s 3 λ 3 Σ ξ 3 w 2 dsdt + C sλ Σ ξ|∇w| 2 dsdt.
By applying now a Young's inequality to B 32 , we obtain

|B 32 | ≤ sλ Σ ξ|∇w| 2 dsdt + 1 sλ Σ 1 ξ |∂ t w| 2 dsdt.
Lastly, since

∂ t φ = (2t -T ) e λ(2L+η
) -e 4λL θ 2 (t) and e 4λL ≤ e 2λ(2L+η) , we have

|∂ t φ| ≤ T ξ 2 .
(2) As a consequence

|B 23 | ≤ s 2 λ Σ T ξ 3 w 2 dsdt
and since s ≥ ρ T , we obtain for large ρ and large λ,

|B 23 | ≤ C s 3 λ 3 Σ ξ 3 w 2 dsdt.

By gathering the above estimates we obtain that

Σ B(w) dxdt ≤ C s 3 λ 3 Σ ξ 3 w 2 dsdt + sλ Σ ξ|∇w| 2 dsdt + 1 sλ Σ 1 ξ (∂ t w) 2 dsdt ,
so that there exist some constants λ 0 , ρ 0 > 0 and C > 0 such that for all T , for all λ > λ 0 , ρ > ρ 0 and all u ∈ C 2 (Q),

s 3 λ 4 Q ξ 3 w 2 dxdt + sλ 2 Q ξ|∇w| 2 dxdt ≤ C Q |P u| 2 e 2sφ dxdt + C s 3 λ 3 Σ ξ 3 w 2 dsdt + C sλ Σ ξ|∇w| 2 dsdt + C 1 sλ Σ 1 ξ (∂ t w) 2 dsdt.
To complete the proof we have to come back to the function u by using w = ue sφ , so that ∇w = (∇u + sλξ∇η u)e sφ , ∂ t w = (∂ t u + s∂ t φ u)e sφ .

There exists some c > 0 such that

s 3 λ 4 Q ξ 3 w 2 dxdt + sλ 2 Q ξ|∇w| 2 dxdt ≥ c s 3 λ 4 Q ξ 3 |∇η| 2 w 2 dxdt + c sλ 2 Q ξ|∇w| 2 dxdt ≥ c s 3 λ 4 Q ξ 3 |∇η| 2 u 2 e 2sφ dxdt + c sλ 2 Q ξ|∇u + sλξ∇η u| 2 e 2sφ dxdt = 2c s 3 λ 4 Q ξ 3 |∇η| 2 u 2 e 2sφ dxdt + c sλ 2 Q ξ|∇u| 2 e 2sφ dxdt + 2c s 2 λ 3 Q ξ 2 (∇u • ∇η)u e 2sφ dxdt.
From the Young's inequality, for any r ∈ (0, 1),

2 s 2 λ 3 Q ξ 2 (∇u • ∇η)ue 2sφ dxdt ≤ 1 r s 3 λ 4 Q ξ 3 |∇η| 2 u 2 e 2sφ dxdt + r sλ 2 Q ξ|∇u| 2 e 2sφ dxdt,
so that

s 3 λ 4 Q ξ 3 w 2 dxdt + sλ 2 Q ξ|∇w| 2 dxdt ≥ c 2 - 1 r s 3 λ 4 Q ξ 3 |∇η| 2 u 2 e 2sφ dxdt + c(1 -r) sλ 2 Q ξ|∇u| 2 e sφ dxdt.
By choosing r ∈ (1/2, 1) we obtain there exists c > 0 such that

s 3 λ 4 Q ξ 3 w 2 dxdt + sλ 2 Q ξ|∇w| 2 dxdt ≥ c s 3 λ 4 Q ξ 3 u 2 e 2sφ dxdt + c sλ 2 Q ξ|∇u| 2 e 2sφ dxdt.
Besides, we have

s 3 λ 3 Σ ξ 3 w 2 dsdt + sλ Σ ξ|∇w| 2 dsdt = s 3 λ 3 Σ ξ 3 u 2 e 2sφ dsdt + sλ Σ ξ|∇u + sλξ∇η u| 2 e 2sφ dsdt ≤ C s 3 λ 3 Σ ξ 3 u 2 e 2sφ dsdt + C sλ Σ ξ|∇u| 2 e 2sφ dsdt and 1 sλ Σ 1 ξ (∂ t w) 2 dsdt = 1 sλ Σ 1 ξ (∂ t u + s∂ t φ u) 2 e 2sφ dsdt ≤ C 1 sλ Σ 1 ξ (∂ t u) 2 e 2sφ dsdt + C s λ Σ 1 ξ (∂ t φ) 2 u 2 e 2sφ dsdt,
which from (2) implies that

1 sλ Σ 1 ξ (∂ t w) 2 dsdt ≤ C 1 sλ Σ 1 ξ (∂ t u) 2 e 2sφ dsdt + C sT 2 λ Σ ξ 3 u 2 e 2sφ dsdt.
From s ≥ ρ T , we obtain for large ρ and large λ,

1 sλ Σ 1 ξ (∂ t w) 2 dsdt ≤ C 1 sλ Σ 1 ξ (∂ t u) 2 e 2sφ dsdt + C s 3 λ 3 Σ ξ 3 u 2 e 2sφ dsdt,
which completes the proof of the Theorem.

4. Proof of the Main Theorem. Let us denote Γ 1 the complementary part of Γ 0 in ∂Ω and Σ 1 = Γ 1 × (0, T ). As a consequence of the previous Carleman estimate, we will obtain an estimate in a subdomain of Q defined as the product of the space open domain Ω ε defined by

Ω ε = {x ∈ Ω, d(x, Γ 1 ) > ε}
by a time interval centered at an arbitrary point in (0, T ) and the length of which is of order ε 3/2 . Lemma 4.1. There exist some positive constants ν, c, C 1 , C 2 , C, ρ 0 , ε 0 such that for all t -, t + with 0 ≤ t -< t + ≤ T , for all ρ > ρ 0 and all ε ∈ (0, ε 0 ), for all u ∈ C 2 (Q),

||u|| 2 L 2 (t ε -,t ε + ;H 1 (Ωε)) ≤ C e 2C1ρ(1+1/τ ) ||P u|| 2 L 2 (q) + ||u|| 2 H 1 (σ0) + ||∂ n u|| 2 L 2 (σ0) + C ε -2ν e -2C2ρε(1+1/τ ) ||u|| 2 L 2 (t-,t+;H 1 (Ω)) , with q = Ω × (t -, t + ), σ 0 = Γ 0 × (t -, t + ), t ε -= t -+ t + 2 -c √ ε t + -t - 2 , t ε + = t -+ t + 2 + c √ ε t + -t - 2 ,
and τ = t + -t -.

Proof. The proof is based on the Carleman estimate of Theorem 3.1 in the domain q = Ω × (t -, t + ). We also denote σ = ∂Ω × (t -, t + ). We recall that either (Γ 0 , Γ 1 ) = (∂O, ∂D) or (Γ 0 , Γ 1 ) = (∂D, ∂O). In order to specify the function η used in the Carleman estimate we have to consider these two cases separately in view of Assumption 1.1: we apply Theorem 2.1 for m = 2

• either with D = D and O = O if (Γ 0 , Γ 1 ) = (∂O, ∂D), • or with D = B \ O and O = B \ D if (Γ 0 , Γ 1 ) = (∂D, ∂O), where B is an open ball of R d such that D B. In both cases we obtain a function η ∈ C 2 (Ω) such that η ≥ 0 in Ω, η > 0 in Ω \ Γ 1 , ∇η = 0 in Ω and η coincides with the distance function to the boundary in a tubular vicinity of Γ 1 . For any u ∈ C 2 (Q), let us define v ε = χ ε u, where χ ε ∈ C 2 (Ω) is a function of x with values in [0, 1] and such that, for sufficiently small ε χ ε (x) = 1 if d(x, Γ 1 ) ≥ 2ε χ ε (x) = 0 if d(x, Γ 1 ) ≤ ε.
We can choose χ ε such that

|∇χ ε (x)| ≤ C ε , |∆χ ε (x)| ≤ C ε 2 , ∀x ∈ Ω. To construct the function χ ε it suffices to define a function f ε ∈ C 2 (R) with values in [0, 1] such that f ε (z) = 1 if z ≥ 2ε f ε (z) = 0 if z ≤ ε, with |f ε (z)| ≤ C/ε and |f ε (z)| ≤ C/ε 2 , then to set χ ε = f ε • d(•, Γ 1 ). Hence v ε ∈ C 2 (Q)
and vanishes in the vicinity of Σ 1 .

We now apply Theorem 3.1 to the function v ε in the domain q = Ω × (t -, t + ), with θ(t) = (t -t -)(t + -t). For a fixed value of λ > λ 0 , let us denote

g(z) = e 4λL -e λ(2L+z) , z ∈ [0, L],
which is a positive and decreasing function.

Let us first consider the left-hand side of the Carleman estimate. For small ε and x ∈ Ω 3ε , we have η(x) ≥ 3ε. Let us define t m = (t -+ t + )/2. For any t r ∈ (t -, t m ), and t ∈ (t r , t -+ t + -t r ), θ(t) ≥ θ(t r ), that is

e sφ(x,t) = e -s g(η(x)) θ(t) ≥ e -s g(3ε) θ(tr ) ,
We have already seen that sξ ≥ 1 for large ρ, hence

s 3 λ 4 t+ t- Ω ξ 3 v 2 ε e 2sφ dxdt + sλ 2 t+ t- Ω ξ|∇v ε | 2 e 2sφ dxdt ≥ e -2s g(3ε) θ(tr ) t++t--tr tr Ω3ε (u 2 + |∇u| 2 ) dxdt.
Let us now consider the right-hand side. We have

P (χ ε u) = χ ε P u -2∇χ ε • ∇u -∆χ ε u. q (P v ε ) 2 e 2sφ dxdt ≤ C t+ t- Ω (P u) 2 e 2sφ dxdt + C ε -4 t+ t- Ωε\Ω2ε (u 2 + |∇u|) 2 e 2sφ dxdt.
For x ∈ Ω, we have 0 ≤ η(x) ≤ L. For t ∈ (t -, t + ), we have θ(t) ≤ θ(t m ), hence For x ∈ Ω ε \ Ω 2ε , we have 0 ≤ η(x) ≤ 2ε. For t ∈ (t -, t + ), we obtain

e sφ(x,t) = e -s g(η(x)) θ(t) ≤ e -s g(2ε) θ(tm) , so that t+ t- Ωε\Ω2ε (u 2 + |∇u| 2 ) e 2sφ dxdt ≤ e -2s g(2ε) θ(tm) t+ t- Ω (u 2 + |∇u| 2 ) dxdt.
We have, since sξ ≥ 1 for large ρ,

s 3 λ 3 σ ξ 3 v 2 ε e 2sφ dsdt + sλ σ ξ|∇v ε | 2 e 2sφ dsdt + 1 sλ σ 1 ξ (∂ t v ε ) 2 e 2sφ dsdt ≤ C σ0 (sξ) 3 (u 2 + |∇u| 2 + (∂ t u) 2 )e -2s g(L) θ(t) dsdt ≤ C σ0 (u 2 + |∇u| 2 + (∂ t u) 2 )f (s/θ(t)) dsdt, with f (z) = z 3 e -2g(L)z .
Here we have used the fact that for fixed λ, |ξ| ≤ C/θ. It is not difficult to see that f (z) ≤ C e -g(L)z for z ≥ 0, so that

s 3 λ 3 σ ξ 3 v 2 ε e 2sφ dsdt + sλ σ ξ|∇v ε | 2 e 2sφ dsdt + 1 sλ σ 1 ξ (∂ t v ε ) 2 e 2sφ dsdt ≤ C e -s g(L) θ(tm) σ0 (u 2 + |∇u| 2 + (∂ t u) 2 ) dsdt.
Gathering the previous estimates, we obtain that for large ρ, e -2s g(3ε)

θ(tr )

t++t--tr tr Ω3ε

(u 2 + |∇u| 2 ) dxdt ≤ C e -2s g(L) θ(tm) t+ t- Ω (P u) 2 dxdt + C ε -4 e -2s g(2ε) θ(tm) t+ t- Ω (u 2 + |∇u| 2 ) dxdt + C e -s g(L) θ(tm ) σ0 (u 2 + |∇u| 2 + (∂ t u) 2 ) dsdt.
We finally obtain that for large ρ, e -2s g(3ε) θ(tr

) ||u|| 2 L 2 (tr,t++t--tr;H 1 (Ω3ε)) ≤ C e -s g(L) θ(tm) ||P u|| 2 L 2 (q) + ||u|| 2 H 1 (σ0) + ||∂ n u|| 2 L 2 (σ0) + C ε -4 e -2s g(2ε) θ(tm ) ||u|| 2 L 2 (t-,t+;H 1 (Ω)) , that is ||u|| 2 L 2 (tr,t++t--tr;H 1 (Ω3ε)) ≤ C e 2s p1 ||P u|| 2 L 2 (q) + ||u|| 2 H 1 (σ0) + ||∂ n u|| 2 L 2 (σ0) + C ε -4 e -2s p2 ||u|| 2 L 2 (t-,t+;H 1 (Ω)) , with p 1 = g(3ε) θ(t r ) - g(L) 2θ(t m ) , p 2 = g(2ε) θ(t m ) - g(3ε) θ(t r ) .
For all γ ∈ (0, 1) there is a unique t r ∈ (t -, t m ) such that θ(t r ) = γ θ(t m ) = γ τ 2 /4. We have

p 1 ≤ g(0) θ(t r ) and p 2 = g(2ε) θ(t r ) γ - g(3ε) g (2ε)
.

The function g(3ε)/g(2ε) is equal to 1 -c 0 ε near ε = 0 at order 1. By choosing γ = 1 -c 1 ε, with 0 < c 1 < c 0 then for small ε there exist some constants C 1 , C 2 > 0 such that

p 1 ≤ C 1 τ 2 , p 2 ≥ C 2 ε τ 2 , for some constants C 1 , C 2 > 0.
The time t r is obtained by solving the equation

(t + -t r )(t r -t -) = γ (t + -t -) 2 4 , that is t r = t -+ t + 2 -β t + -t - 2 , β = 1 -γ = √ c 1 √ ε,
which is the result by setting c = c 1 /3 and setting ε = 3ε.

Then we gather the estimates obtained in Lemma 4.1 for a sequence of time intervals in order to obtain an estimate in the subdomain Ω ε × (ε, T -ε) of Q.

Proposition 1. There exist some positive constants ν, C 1 , C 2 , C, ρ 0 , ε 0 such that for all ρ > ρ 0 and all ε ∈ (0, ε 0 ), for all u ∈ C 2 (Q),

||u|| L 2 (ε,T -ε;H 1 (Ωε)) ≤ C e C1ρ/ε ||P u|| L 2 (Q) + ||u|| H 1 (Σ0) + ||∂ n u|| L 2 (Σ0) + C ε -ν e -C2ρ ||u|| L 2 (0,T ;H 1 (Ω)) .
Proof. Let us take γ = 1 -c 2 ε, where c is as in the statement of Lemma 4.1,

β = √ 1 -γ = c √ ε and some τ > 0. Let us define, for n ∈ N such that τ (1 + nβ) ≤ T , the sequences t n -= nβτ t n + = τ + nβτ and      tn -= τ 2 + (n - 1 2 )βτ tn + = τ 2 + (n + 1 2 )βτ.
By using the notations of Lemma 4.1, for t -= t n -and t + = t n + , the corresponding values of t ε -and t ε + are

t ε -= t n -+ t n + 2 -c √ ε t n + -t n - 2 , t ε -= τ 2 + nβτ -β τ 2 = tn - and t ε + = t n -+ t n + 2 + c √ ε t n + -t n - 2 , t ε + = τ 2 + nβτ + β τ 2 = tn + .
Applying Lemma 4.1 we obtain that for large ρ and small ε, for all u ∈ C 2 (Q),

||u|| 2 L 2 ( tn -, tn + ;H 1 (Ωε)) ≤ C e 2C1ρ(1+1/τ ) ||P u|| 2 L 2 (t n -,t n + ;L 2 (Ω)) + ||u|| 2 H 1 (σ n 0 ) + ||∂ n u|| 2 L 2 (σ n 0 ) + C ε -2ν e -2C2ρε(1+1/τ ) ||u|| 2 L 2 (t n -,t n + ;H 1 (Ω)) , where σ n 0 = Γ 0 × (t n -, t n + ). Let us consider N such that τ (1 + N β) ≤ T . We have N n=0 ||u|| 2 L 2 ( tn -, tn + ;H 1 (Ωε)) ≤ C(N + 1) e 2C1ρ(1+1/τ ) ||P u|| 2 L 2 (Q) + ||u|| 2 H 1 (Σ0) + ||∂ n u|| 2 L 2 (Σ0) + C(N + 1) ε -2ν e -2C2ρε(1+1/τ ) ||u|| 2 L 2 (0,T ;H 1 (Ω)) .
It is readily seen that tn + = tn+1 -, we hence end up with

||u|| 2 L 2 ( t0 -, tN + ;H 1 (Ωε)) ≤ C(N + 1) e 2C1ρ(1+1/τ ) ||P u|| 2 L 2 (Q) + ||u|| 2 H 1 (Σ0) + ||∂ n u|| 2 L 2 (Σ0) + C(N + 1) ε -2ν e -2C2ρε(1+1/τ ) ||u|| 2 L 2 (0,T ;H 1 (Ω)) . Since t0 -= τ 2 (1 -β), tN + = τ 2 + (N + 1 2 )βτ, by choosing τ such that τ (1 -β)/2 = ε, that is τ = 2ε 1 -c √ ε
and then N such that

τ (1 + N β) ≤ T, τ 2 + (N + 1 2 )βτ ≥ T 2 , ||u|| 2 L 2 (ε,T /2;H 1 (Ωε)) ≤ C(N + 1) e 2C1ρ(1+1/τ ) ||P u|| 2 L 2 (0,T ;L 2 (Ω)) + ||u|| 2 H 1 (Σ0) + ||∂ n u|| 2 L 2 (Σ0) + C(N + 1) ε -2ν e -2C2ρε(1+1/τ ) ||u|| 2 L 2 (0,T ;H 1 (Ω))
. We can make a symmetric construction with respect to t = T /2. In addition, we have τ βN ≤ T , hence there exists a constant C such that

N ≤ C ε 3/2 .
We conclude that there exist some positive constants ν, C 1 , C 2 , C such that for large ρ and small ε, for all u ∈ C 2 (Q),

||u|| 2 L 2 (ε,T -ε;H 1 (Ωε)) ≤ C e 2C1ρ/ε ||P u|| 2 L 2 (Q) + ||u|| 2 H 1 (Σ0) + ||∂ n u|| 2 L 2 (Σ0) + C ε -2ν e -2C2ρ ||u|| 2 L 2 (0,T ;H 1 (Ω))
, which completes the proof. Remark 2. If for a fixed arbitrary ε we optimize ρ in the estimate of Proposition 1, we obtain a standard Hölder stability estimate in Ω ε × (ε, T -ε) as in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF].

In what follows we will need the following Hardy-type inequality, that is Lemma 4.2. If Q ⊂ R d is a bounded, connected and Lipschitz domain, and if d ∂Q (x) denotes the distance of x to ∂Q, then ∀r ∈ (0, 1/2), ∀u ∈ H r (Q),

u d r ∂Q L 2 (Q) ≤ C ||u|| H r (Q) ,
with C > 0 depending only on r and on Q. Lemma 4.2 is for example proved in [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. Next we obtain the following result, which combines Proposition 1 and Lemma 4.2. More precisely, the Hardy-type inequality of Lemma 4.2 enables us to extend the estimate of Proposition 1 in the subdomain Ω ε × (ε, T -ε) to the whole domain Q = Ω × (0, T ). Proposition 2. For all s ∈ (0, 1), there exist some constants c, ε 0 > 0 such that for all ε ∈ (0, ε 0 ), for all u ∈ H 1 (0, T ;

H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), ||u|| L 2 (0,T ;H 1 (Ω)) ≤ e c/ε ||P u|| L 2 (Q) + ||u|| H 1 (Σ0) + ||∂ n u|| L 2 (Σ0) + ε s (||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) ).
Proof. We first specify ρ as a function of ε. For some r ∈ (0, 1), we set

ε -ν e -C2ρ = ε r , which implies that e C1 ρ/ε = 1 ε (r+ν)/C2 C1/ε ≤ e (c/ε) log(1/ε) ,
for some c > 0. By the Proposition 1 we conclude that for small ε, for all u ∈ C 2 (Q),

||u|| L 2 (ε,T -ε;H 1 (Ωε)) ≤ C e (c/ε) log(1/ε) ||P u|| L 2 (Q) + ||u|| H 1 (Σ0) + ||∂ n u|| L 2 (Σ0) + C ε r ||u|| L 2 (0,T ;H 1 (Ω)) . (3)
Now let us use Lemma 4.2 in the (d+1)-dimensional domain Q = Ω × (0, T ). We have for all r ∈ (0, 1),

u/d r/2 ∂Q L 2 (Q) ≤ C ||u|| H r/2 (Q) ≤ C ||u|| H 1/2 (Q) .
In particular, by denoting

Q ε = Ω ε × (ε, T -ε) and R ε = Q \ Q ε , since d ∂Q (x, t) ≤ ε for all (x, t) ∈ R ε , we obtain ||u|| L 2 (Rε) ≤ C ε r/2 ||u|| H 1/2 (Q) ≤ C ε r/2 ||u|| 1/2 L 2 (Q) ||u|| 1/2 H 1 (Q) , which implies that for any η > 0, ||u|| L 2 (Rε) ≤ C η ||u|| L 2 (Q) + ε r η ||u|| H 1 (Q) .
We notice that the domain R ε is the union of three cylindrical domains, namely

R ε = Ω ε × (0, ε) ∪ Ω ε × (T -ε, T ) ∪ ω ε × (0, T ),
where ω ε = Ω \ Ω ε , so that by using the previous inequality for functions u and its first derivatives with respect to x i , i = 1, • • • , d, we end up with

||u|| L 2 (0,ε;H 1 (Ωε)) + ||u|| L 2 (T -ε,T ;H 1 (Ωε)) + ||u|| L 2 (0,T ;H 1 (ωε)) ≤ C η ||u|| L 2 (0,T ;H 1 (Ω)) + ε r η (||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) ). ( 4
)
Using the fact that

||u|| L 2 (0,T ;H 1 (Ω)) ≤ ||u|| L 2 (ε,T -ε;H 1 (Ωε)) + ||u|| L 2 (0,ε;H 1 (Ωε)) + ||u|| L 2 (T -ε,T ;H 1 (Ωε)) + ||u|| L 2 (0,T ;H 1 (ωε)) ,
by gathering the estimates ( 3) and ( 4) and by choosing η such that C η = 1/2, we end up with

||u|| L 2 (0,T ;H 1 (Ω)) ≤ C e (c/ε) log(1/ε) (||P u|| L 2 (Q) + ||u|| H 1 (Σ0) + ||∂ n u|| L 2 (Σ0) ) + C ε r (||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) ).
Now for any a > 1 we can choose c such that for small ε, e (c/ε) log(1/ε) ≤ e c /ε a .

Defining ε = ε a , we have that ε r = ε s with s = r/a. Since for any s ∈ (0, 1) we can choose some r ∈ (0, 1) and a > 1 such that s = r/a, this implies the result for any function in C 2 (Q). We conclude by density of

C ∞ (Q) in H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)).
We can now give the proof of the Main Theorem, which essentially consists of a classical optimization process in the estimate of Proposition 2 with respect to the small parameter ε.

Proof of the Main Theorem. Let us pick s ∈ (0, 1). By denoting

δ = ||P u|| L 2 (0,T ;L 2 (Ω)) + ||u|| H 1 (Σ0) + ||∂ n u|| L 2 (Σ0) , m = ||u|| L 2 (0,T ;H 1 (Ω)) and M = ||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) , we have that for ε ∈ (0, ε 0 ), m ≤ e c/ε δ + ε s M. ( 5
)
Here c and ε 0 only depend on s. Denoting f (ε) = e c/ε δ + ε s M for ε > 0, the minimizer ε m of f solves

g(ε m ) = M δ , g(ε) := c s 1 ε s+1 e c/ε .
The function g is decreasing with g(0 + ) = +∞ and g(+∞) = 0, so that the above equation has a unique solution ε m for each M, δ > 0. We treat two cases separately.

• If ε 0 > ε m , then by choosing ε = ε m in ( 5) we obtain that

m ≤ ( s c ε 0 + 1)M ε s m = CM ε s m . ( 6
)
For sufficiently large M/δ, ε m is sufficiently small to have for some c > c,

M δ = g(ε m ) ≤ e c /εm .
It follows that ε m ≤ c / log(M/δ), and by plugging this estimate in ( 6) we obtain that for sufficiently large M/δ m ≤ C M log s (M/δ) .

• If ε 0 ≤ ε m , we obtain g(ε 0 ) ≥ M/δ, and thus

m ≤ M ≤ g(ε 0 )δ = C M M/δ .
For large M/δ we have M/δ ≥ log s (M/δ), which implies again [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF]. We conclude that the estimate ( 7) is true for all u such that M/δ is sufficiently large. This implies, up to a change of constant C, that

m ≤ C M log s (2 + M/δ)
as soon as M/δ is sufficiently large. But since m ≤ M , such an estimate is true for any M/δ, up to a change of constant C. Remark 3. Another presentation of the Main Theorem is the following: for all s ∈ (0, 1), there exists a constant C > 0 such that for all M, δ > 0, for all u ∈ H 1 (0, T ;

H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) such that ||u|| H 1 (0,T ;H 1 (Ω)) +||u|| L 2 (0,T ;H 2 (Ω)) ≤ M, ||P u|| L 2 (Q) +||u|| H 1 (Σ0) +||∂ n u|| L 2 (Σ0) ≤ δ, then ||u|| L 2 (0,T ;H 1 (Ω)) ≤ C M log s (2 + M/δ) .
Remark 4. The Main Theorem can be extended with no additional difficulty to the case when the operator P = ∂ t -∆ is replaced by

P = ∂ t -div(σ∇) + B • ∇ + a with σ ∈ W 1,∞ (Ω), B ∈ (L ∞ (Ω)) d , a ∈ L ∞ (Ω) and σ(x) ≥ c > 0 for all x ∈ Ω.
Remark 5. It is readily seen that for a stationary solution u, that is when u is independent of time t, the Main Theorem reduces to: for all s ∈ (0, 1), there exists a constant C > 0 such that for all u ∈ H 2 (Ω),

||u|| H 1 (Ω) ≤ C ||u|| H 2 (Ω) log s 2 + ||u|| H 2 (Ω) ||∆u|| L 2 (Ω) +||u|| H 1 (Γ 0 ) +||∂nu|| L 2 (Γ 0 )
, which is the result already obtained in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de Laplace[END_REF][START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 , M2AN Math[END_REF] for the Laplace operator. Hence our Main Theorem can be seen as a generalization of the result in the stationary case. In addition, under the restriction of Assumption 1.1, such result for the Laplacian could be obtained more easily than in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de Laplace[END_REF][START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 , M2AN Math[END_REF] by using a Carleman weight independent of time and based on the same function η(x) as in the proof of the non stationary case.

5. Optimality of our stability estimate. In this section we discuss the optimality of the estimate given by the Main Theorem. For sake of simplicity we consider the 1D case and a particular 2D case. They correspond to the following geometries.

• In the 1D case, Q = (0, 1)

× (0, T ), Σ 0 = {0} × (0, T ), Σ 1 = {1} × (0, T ). • In the 2D case, Q = (B(O, 2) \ B(O, 1)) × (0, T ), Σ 0 = C(O, 1) × (0, T ), Σ 1 = C(O, 2) × (0, T ), where B(O, r) (resp. C(O, r))
is the open ball (resp. the circle) of center O and radius r, with r = 1, 2. We have the following result, which proves that our logarithmic stability estimate is optimal, up to the exponent of the log. More precisely, we have proved in the Main Theorem that our estimate holds with exponent s = 1 -ε, while we establish now that this exponent cannot be larger than 2 in 1D and cannot be larger than 1 in 2D.

Theorem 5.1. Let us consider one of the 1D or 2D geometries above. Assume there exists a non decreasing function f : R + → R + such that for all u ∈ H 1 (0, T ;

H 1 (Ω))∩ L 2 (0, T ; H 2 (Ω)), ||u|| L 2 (0,T ;H 1 (Ω)) ≤ ||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) f ||u|| H 1 (0,T ;H 1 (Ω)) +||u|| L 2 (0,T ;H 2 (Ω)) ||P u|| L 2 (Q) +||u|| H 1 (Σ 0 ) +||∂nu|| L 2 (Σ 0 )
. Then there exists some constant C such that for large A,

f (A) ≤ C log s (A),
where s = 2 in 1D and s = 1 in 2D.

Proof. We first focus on the 1D case. Let us consider for all a > 0 the smooth functions defined on [0, 1] × [0, T ] by

u a (x, t) = χ(x)v a (x, t), v a (x, t) = e ax+a 2 t ,
where χ is a smooth function on the segment [0, 1] such that χ(0) = χ (0) = 0, χ(x) ∈ [0, 1] for all x ∈ [0, 1] and χ(x) = 1 for all x ∈ [X, 1] for some X ∈ (0, 1). Let us now denote

m a = ||u a || L 2 (0,T ;H 1 (0,1)) , M a = ||u a || H 1 (0,T ;H 1 (0,1)) + ||u a || L 2 (0,T ;H 2 (0,1)) and δ a = ||P u a || L 2 (Q) + ||u a || H 1 (Σ0) + ||∂ x u a || L 2 (Σ0) .
We hence have, for all a > 0,

m a ≤ M a f (M a /δ a ) . (8)
It is important to note that the functions v a satisfy the heat equation P v a = 0 in (0, 1) × (0, T ) and that, due to the boundary conditions satisfied by χ at x = 0, we have u a = 0 and ∂ x u a = 0 on Σ 0 . We hence have

δ a = ||P u a || L 2 (Q) with P u a = -2χ ∂ x v a -χ v a .
The idea is now to minorate the left-hand side of ( 8) and to majorate the right-hand side. As concerns the left-hand side we have

m 2 a = T 0 1 0 (u 2 a + (∂ x u a ) 2 ) dxdt ≥ T 0 1 X (v 2 a + (∂ x v a ) 2 ) dxdt = T 0 1 X (1 + a 2 )e 2ax+2a 2 t dxdt = (1 + a 2 ) e 2a 2 T -1 2a 2
e 2a -e 2aX 2a Hence there exists some constant c > 0 such that for large a, m a ≥ c a -1/2 e a 2 T +a .

Now let us consider the right-hand side.

M 2 a = T 0 1 0 u 2 a + (∂ x u a ) 2 + (∂ t u a ) 2 + (∂ 2 xt u a ) 2 + (∂ 2 xx u a ) 2 dxdt.
There exists a constant C such that for large a, Hence there exists some constant C > 0 such that for large a, M a ≤ C a 3/2 e a 2 T +a .

On the other hand, we have

M 2 a ≥ T 0 1 X v 2 a + (∂ x v a ) 2 + (∂ t v a ) 2 + (∂ 2 xt v a ) 2 + (∂ 2 xx v a ) 2 dxdt.
We conclude as previously that there exists a constant c such that for large a, M a ≥ c a 3/2 e a 2 T +a .

Lastly, there exists some constant C such that

δ 2 a ≤ C T 0 X 0 v 2 a + (∂ x v a ) 2 dxdt.
We conclude that there exists some constant C such that for large a, δ a ≤ Ca -1/2 e a 2 T +aX .

Plugging the estimates ( 9), [START_REF] Phung | Remarques sur l'observabilité pour l'équation de Laplace[END_REF], [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 , M2AN Math[END_REF] and [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF] in estimate (8) and using the fact that f in non-decreasing, we obtain that there exist some constants c, C such that for large a, f (c a 2 e a(1-X) ) ≤ C a 2 . We complete the proof by setting A = c a 2 e a(1-X) . Now let us address the 2D case, for which we have supposed that Ω = B(O, 2) \ B(O, 1). Let us consider for all p ∈ N, p = 0, the smooth functions defined on Ω × [0, T ] by u p (x, t) = χ(r)v p (x, t), v p (x, t) = r p sin(p θ), where (r, θ) are the polar coordinates and χ is a smooth function of r on the segment [START_REF] Bourgeois | The "exterior approach" applied to the inverse obstacle problem for the heat equation[END_REF][START_REF] Bécache | Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case[END_REF] such that χ(1) = χ (1) = 0, χ(r) ∈ [0, 1] for all r ∈ [1, 2] and χ(r) = 1 for all r ∈ [R, 2] for some R ∈ (1, 2). Note that the functions u p and v p are independent of time t and that u p = 0 and ∂ r u p = 0 on the inner boundary C(O, 1), so that we have for all p

m p ≤ M p f (M p /δ p ) , (13)
with

m p = ||u p || H 1 (Ω) , M p = ||u p || H 2 (Ω) , δ p = ||∆u p || L 2 (Ω) .
By repeating the same computations as for the 1D case, we prove that there exist two constants c, C > 0 such that for large p,

m p ≥ c p 1/2 2 p , c p 3/2 2 p ≤ M p ≤ C p 3/2 2 p , δ p ≤ C p 1/2 R p . (14)
As an example of such computation, the minoration of m p is obtained as follows:

m 2 p = 2 1 2π 0 u 2 p + (∂ r u p ) 2 + 1 r 2 (∂ θ u p ) 2 r drdθ ≥ 2 R 2π 0 v 2 p + (∂ r v p ) 2 + 1 r 2 (∂ θ v p ) 2 r drdθ = 2 R 2π 0 r 2p+1 sin 2 (p θ) + p 2 r 2p-1 drdθ = π 2p + 2 (2 2p+2 -R 2p+2 ) + πp (2 2p -R 2p ),
which implies the result. To obtain the majoration of δ p it is useful to remark that ∆v p = 0 for all p. By plugging the estimates ( 14) in ( 13), we conclude that there exist some constants c, C > 0 such that for large p,

f (c p (2/R) p ) ≤ C p,
and we conclude as in the 1D case.

Remark 6. We hence prove that for d = 2, the estimate for s = 1 -ε of our Main Theorem is quasi-optimal, in the sense that the result given by Theorem 5.1 contradicts the validity of the estimate for exponent s > 1, but not for s = 1. We can probably adapt the example for d = 2 to prove such quasi-optimality for any dimension d > 2.

6. An application to the identification of initial condition. In this section we come back to the general case of dimension d with Assumption 1.1. As an application of the Main Theorem, let us show that the initial condition, namely u 0 := u| S0 for S 0 = Ω × {0}, can be estimated from the data (f, g 0 , g 1 ) in problem [START_REF] Bourgeois | The "exterior approach" applied to the inverse obstacle problem for the heat equation[END_REF]. More precisely, we have the following result.

Corollary 1. For all s ∈ (0, 1), there exists a constant C > 0 such that for all u ∈ H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)),

||u 0 || L 2 (Ω) ≤ C M (u)   1 log s 2 + M (u) δ(u) + δ(u) M (u)   3 4
,

where δ(u) = ||P u|| L 2 (Q) + ||u|| H 1 (Σ0) + ||∂ n u|| L 2 (Σ0) , M (u) = ||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) .
Proof. Given some u ∈ H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), it is useful to introduce for (x, t) ∈ Q the vector field U (x, t) ∈ R d+1 defined by

U = (∇u, -u) = (∂ xi u, -u), i = 1, • • • , d.
Clearly, div d+1 U = ∆u -∂ t u = -P u, so that by defining

H div(Q) = {U ∈ (L 2 (Q)) d+1 , div d+1 U ∈ L 2 (Q)}, we have U ∈ H div(Q) and ||U || 2 H div(Q) = ||u|| 2 L 2 (0,T ;H 1 (Ω)) + ||P u|| 2 L 2 (Q) . From a standard trace result in the Lipschitz domain Q, there exists a constant C > 0 such that for all U ∈ H div(Q) , ||U • n d+1 || H -1/2 (∂Q) ≤ C ||U || H div(Q) ,
where n d+1 is the outward unit normal to the domain Q. This implies in particular that u 0 = u| S0 ∈ H -1/2 (Ω) and there exists a constant C > 0 such that for all u ∈ H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)),

||u 0 || H -1/2 (Ω) ≤ C (||u|| L 2 (0,T ;H 1 (Ω)) + ||P u|| L 2 (Q) ),
where H -1/2 (Ω) is the space of restrictions to Ω of distributions in H -1/2 (R d ), that is by using the Main Theorem,

||u 0 || H -1/2 (Ω) ≤ C   M (u) log s 2 + M (u) δ(u) + δ(u)   . (15)
We need an additional estimate for u 0 in a stronger norm. That u ∈ H 1 (0, T ;

H 1 (Ω)) ⊂ C 0 ([0, T ]; H 1 (Ω)) immediately implies that u 0 ∈ H 1 (Ω)
. But furthermore we can prove by using the second assumption u ∈ L 2 (0, T ;

H 2 (Ω)) that u 0 ∈ H 3/2 (Ω). Indeed we remark that U ∈ (H 1 (Q)) d+1 and ||U || 2 (H 1 (Q)) d+1 = ||u|| 2 H 1 (0,T ;H 1 (Ω)) + ||u|| 2 L 2 (0,T ;H 2 (Ω)) .
From a second standard trace result in the domain Q, we have for all U ∈ (H

1 (Q)) d+1 , ||U || (H 1/2 (∂Q)) d+1 ≤ C ||U || (H 1 (Q)) d+1 ,
which implies in particular, given the definition of U , that the trace on S 0 of the partial derivatives

∂ xi u, i = 1, • • • , d, belongs to H 1/2 (Ω). By density of C ∞ (Q) in H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), we clearly have (∂ xi u)| S0 = ∂ xi (u| S0 ), i = 1, • • • , d, that is ∂ xi u 0 ∈ H 1/2 (Ω).
We finally end up with u 0 ∈ H 3/2 (Ω) and there exists a constant C > 0 such that for all u ∈ H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)),

||u 0 || H 3/2 (Ω) ≤ C (||u|| H 1 (0,T ;H 1 (Ω)) + ||u|| L 2 (0,T ;H 2 (Ω)) ), that is ||u 0 || H 3/2 (Ω) ≤ C M (u). ( 16
) By an interpolation result of [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] (see Remark 12.6 applied to s 1 = 1/2, s 2 = 3/2 and θ = 1/4), we have

H 1/2 00 (Ω), (H 3/2 (Ω)) 1/4 = L 2 (Ω),
where H 1/2 00 (Ω) is the subspace of functions in H 1/2 (Ω) such that their extension by 0 to the whole space R d belongs to H 1/2 (R d ) and (H 3/2 (Ω)) is the dual space of H 3/2 (Ω). Since the dual space of H 1/2 00 (Ω) coincides with space H -1/2 (Ω), by using Theorem 6.2 of [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] we obtain

H 3/2 (Ω), H -1/2 (Ω) 3/4 = L 2 (Ω),
as well as the interpolation inequality: there exists a constant C > 0 such that for all u ∈ H 1 (0, T ; H

1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), ||u 0 || L 2 (Ω) ≤ C ||u 0 || 1/4 H 3/2 (Ω) ||u 0 || 3/4 H -1/2 (Ω) .
The conclusion follows from the estimates ( 15) and ( 16).

7. An application to quasi-reversibility. Again we consider the general case of dimension d with Assumption 1.1. The method of quasi-reversibility, first introduced in [START_REF] Lattès | Méthode de quasi-réversibilité et applications[END_REF], is a regularization method that enables us to approximate the solution to the ill-posed problem (1) from the data (f, g 0 , g 1 ) when such a solution exists (it is then unique). Let us assume that f ∈ L 2 (Q), g 0 ∈ H 3/2 (Σ 0 ) and

g 1 ∈ H 1/2 (Σ 0 ), while u ∈ H 2 (Q) satisfies    P u = f in Q u = g 0 on Σ 0 ∂ n u = g 1 on Σ 0 (17)
with P u = ∂ t u -∆u. Let us introduce the subspace V of functions v ∈ H 2 (Q) such that v| Σ0 = 0 and ∂ n v| Σ0 = 0. For α > 0, the quasi-reversibility solution u α associated with any data ( f , g0 , g1 )

∈ L 2 (Q) × H 3/2 (Σ 0 ) × H 1/2 (Σ 0 ) is the unique solution to the weak formulation: find u α ∈ H 2 (Q) such that    (P u α , P v) L 2 (Q) + α(u α , v) H 2 (Q) = ( f , P v) L 2 (Q) for all v ∈ V u α = g0 on Σ 0 ∂ n u α = g1 on Σ 0 . (18)
The unique solvability of problem [START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF] relies on Lax-Milgram's Lemma once we have introduced some U ∈ H 2 (Q) such that U | Σ0 = g0 and ∂ n U | Σ0 = g1 , which is possible in virtue of an extension theorem of [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF], as well as the auxiliary unknown ûα = u α -U . Furthermore, if the data ( f , g0 , g1 ) coincides with the exact data (f, g 0 , g 1 ), it is easy to prove the estimates

||u α -u|| H 2 (Q) ≤ ||u|| H 2 (Q) , ||P (u α -u)|| L 2 (Q) ≤ √ α ||u|| H 2 (Q) , ∀α > 0, ( 19
)
as well as the fact that u α → u in H 2 (Q) when α → 0 (see for example [START_REF] Lattès | Méthode de quasi-réversibilité et applications[END_REF]). A more difficult question is to specify the convergence rate. The first step in this direction was done by M.V. Klibanov (see for example [START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF]), more precisely some Hölder-type convergence rates were obtained in the truncated domain Ω ε × (ε, T -ε). However it is desirable to obtain a convergence rate in the whole domain Q = Ω × (0, T ) as it was done for the Laplacian in [START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 , M2AN Math[END_REF][START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains[END_REF]. Our previous results enable us to derive a logarithmic convergence rate in the whole domain.

Corollary 2. For all s ∈ (0, 1), there exists a constant C > 0 such that for all u ∈ H 2 (Q) satisfying [START_REF] Gilbarg | Elliptic Partial Differential Equations od Second Order[END_REF] for data (f 0 , g 0 , g 1 ) and small α > 0,

||u α -u|| L 2 (0,T ;H 1 (Ω)) ≤ C ||u|| H 2 (Q) log s (1/α) ,
where u α is the solution to problem [START_REF] Rousseau | Applications to unique continuation and control of parabolic equations[END_REF] associated with the data (f 0 , g 0 , g 1 ).

Proof. To obtain the result, we simply use the estimates [START_REF] Fursikov | Controllability of evolution equations[END_REF], the continuous embedding H 2 (Q) ⊂ H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) and the Main Theorem (in the form given in Remark 3) applied to the function (u α -u) with M = ||u|| H 2 (Q) and δ = √ α ||u|| H 2 (Q) .

Remark 7. In the above corollary, the solutions u α , u seem surprisingly smooth. There are two reasons. Firstly, the well-posedness of the weak formulation of quasireversibility (18) requires a lifting in Q of the boundary data (g 0 , g1 ) on Σ 0 , which have to be sufficiently smooth. Secondly, the Main Theorem requires the solutions to be in H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)). For example, following the notations of [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF], it would be interesting and more natural to consider the less regular case when f ∈ L 2 (Q), g0 ∈ H 3/2,3/4 (Σ 0 ), g1 ∈ H 1/2,1/4 (Σ 0 ) and solutions u α , u in H 2,1 (Q). Those boundary data (g 0 , g1 ) can be lifted in H 2,1 (Q) but unfortunately, the space H 1 (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) does not include H 2,1 (Q).

8. About Assumption 1.1. As can be seen in the proof of our Main Theorem, it was possible in a single step to estimate the solution in the subdomain Ω ε ×(ε, T -ε) for small ε > 0 with constants that explicitly depend on ε because of the geometric assumption 1.1. Such assumption implies in particular that Γ 0 ∩ Γ 1 = ∅, where Γ 1 is the complementary part of Γ 0 in ∂Ω and then enabled us to use a smooth spatial function η in the Carleman weight that coincides near the outer or inner boundary with the distance function to the boundary. Such general strategy is no more applicable without Assumption 1.1. However, it is a natural question whether our Main Theorem can be extended to the case when Γ 0 ∩ Γ 1 = ∅. The result given in Remark 5 for the Laplacian in Ω is obtained in [START_REF] Phung | Remarques sur l'observabilité pour l'équation de Laplace[END_REF][START_REF] Bourgeois | About stability and regularization of ill-posed elliptic Cauchy problems: the case of C 1,1 , M2AN Math[END_REF] without Assuming 1.1, by using a more complicated strategy. More precisely, a three steps strategy is used: the smallness of the function is propagated firstly from Γ 0 to the interior of Ω, secondly inside Ω from an open domain to another and thirdly from inside Ω up to the boundary Γ 1 . Such strategy is probably feasible in the case of the heat equation, but the main difficulty is, for each of these three steps, to estimate the constants uniformly with respect to ε. This is a challenging future work.

Assumption 1 . 1 .

 11 The domain Ω satisfies Ω = D \O, where D is an open, bounded and connected domain of class C 2 and O D is an open domain of class C 2 (not necessarily connected), while Γ 0 is either the interior boundary ∂O or the exterior boundary ∂D of Ω.

Theorem 2 . 1 .

 21 Let D ⊂ R d be an open bounded and connected domain of class C m , with m ≥ 2. Let O be an open subset of D. There exists a function η ∈ C m (D) such that • η(x) > 0 for all x ∈ D, • η coincides with the distance function d(•, ∂D) in V = N ∩ D, where N is an open neighborhood of ∂D in R d , • |∇η(x)| > 0 for all x ∈ D \ O.

Lemma 2 . 2 .

 22 Let D ⊂ R d be an open bounded and connected domain of class C m , with m ≥ 2. There exists a function v ∈ C m (D) such that • v(x) > 0 for all x ∈ D, • v coincides with the distance function d(•, ∂D) in V = N ∩ D, where N is an open neighborhood of ∂D in R d , • |∇v(x)| > 0 for all x ∈ V.

3 .

 3 A uniform Carleman estimate. For an open connected domain Ω ⊂ R d of class C 2 , let us consider a function η ∈ C 2 (Ω) such that η(x) ≥ 0 and |∇η(x)| = 0 for x ∈ Ω. Let us define φ(x, t) = e λ(2L+η(x)

e

  sφ(x,t) = e -s g(η(x)) ) 2 e 2sφ dxdt ≤ e -2s g(L) θ(tm ) t+ t-Ω (P u) 2 dxdt.

e 2ax+2a 2 t 2 e

 22 dxdt = Ca 6 e 2a 2 T -1 2a 2a -1 2a .

  = ∂ t w-∆w+2sλξ ∇η•∇w+ -s∂ t φ -s 2 λ 2 ξ 2 |∇η| 2 + sλ 2 ξ|∇η| 2 + sλξ ∆η w, which we write M 1 w + M 2 w = e sφ P u + M 3 w, with M 1 w = 2sλ 2 ξ|∇η| 2 w + 2sλξ∇η • ∇w + ∂ t w, M 2 w = -s 2 λ 2 ξ 2 |∇η| 2 w -∆w -s∂ t φ wand M 3 w = sλ 2 ξ|∇η| 2 w -sλξ ∆η w.

	sφ ,
	while
	∆u = (∆w + s 2 |∇φ| 2 w -2s ∇φ • ∇w -s w ∆φ)e -sφ .
	Given that
	∇φ = λξ∇η, ∆φ = λ 2 ξ 2 |∇η| 2 + λξ ∆η,
	we have
	e sφ P u

Acknowledgments. The author is indebted to Sylvain Ervedoza for his very useful suggestions.


Nicolas Hegoburu
email: nicolas.hegoburu@math.u-bordeaux.fr

Marius Tucsnak
email: marius.tucsnak@math.u-bordeaux.fr

Null controllability of the Lotka-McKendrick system with spatial diffusion

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

Introduction

We consider a linear controlled age-structured population model with spatial diffusion described by the following system:

                    
∂ t p(t, a, x) + ∂ a p(t, a, x) + µ(a)p(t, a, x) -∆p(t, a, x) = χ ω (x)u(t, a, x), t > 0, a ∈ (0, a † ), x ∈ Ω, ∂p ∂ν (t, a, x) = 0, t > 0, a ∈ (0, a † ), x ∈ ∂Ω, p(t, 0, x) = a † 0 β(a)p(t, a, x) da, t > 0, x ∈ Ω, p(0, a, x) = p 0 (a, x), a ∈ (0, a † ), x ∈ Ω.

(1.1)

In the above equations:

• Ω ⊂ R n , n 1, denotes a smooth connected bounded domain and ∆ is the laplacian with respect to the variable x;

• ∂ ∂ν denotes the derivation operator in the direction of the unit outer normal to ∂Ω. We thus have homogeneous Neumann boundary conditions, thus the considered population is isolated from the exterior of Ω;

• p(t, a, x) denotes the distribution density of the population at time t, of age a at spatial position x ∈ Ω;

• p 0 denotes the initial population distribution;

• a † ∈ (0, +∞) is the maximal life expectancy;

• β(a) and µ(a) are positive functions denoting respectively the birth and death rates, which are supposed to be independent of t and x;

• ω ⊂ Ω is a nonempty open susbet of Ω and χ ω denotes the characteristic function of ω.

We make the following classical assumptions on β and µ:

(H1) β ∈ L ∞ (0, a † ), β(a) 0 for almost every a ∈ (0, a † ), (H2) µ ∈ L 1 [0, a * ] for every a * ∈ (0, a † ), µ(a) 0 for almost every a ∈ (0, a † ), (H3)

a † 0 µ(a) da = +∞.
We also introduce the function

π(a) := exp - a 0 µ(s) ds , (1.2)
which is the probability of survival of an individual from age 0 to a.

Our main result is

Theorem 1.1. With the above notations and assumptions, for every τ > 0 and for every p 0 ∈ L 2 ((0, a † )×Ω), there exists u ∈ L ∞ ((0, τ ); L 2 ((0, a † )×ω)) such that the solution p of (1.1) satisfies p(τ, a, x) = 0 (a ∈ (0, a † ), x ∈ Ω a.e.).

(1.

3)

The null-controllability of the the system modelling age-dependant population dynamics is by now well understood in the case in which diffusion is neglected (see Barbu, Ianelli and Martcheva [START_REF] Barbu | On the controllability of the Lotka-McKendrick model of population dynamics[END_REF] and Hegoburu, Magal, Tucsnak [START_REF] Hegoburu | Controllability with positivity constraints of the Lotka-McKendrick system[END_REF]). In the case when spatial diffusion is taken into account, namely for (1.1), the particular case when the control acts in the whole space (the case corresponding to ω = Ω) was investigated by S. Anit ¸a (see [START_REF] Anit ¸a | Analysis and control of age-dependent population dynamics[END_REF], p 148). The case when the control acts in a spatial subdomain ω was firstly studied by B. Ainseba [START_REF] Ainseba | Exact and approximate controllability of the age and space population dynamics structured model[END_REF], where the author proves the null controllability of the above system (1.1), except for a small interval of ages near zero. The case when the control acts in a spatial subdomain ω and also only for small age classes was investigated by B. Ainseba and S. Anit ¸a [START_REF] Ainseba | Internal exact controllability of the linear population dynamics with diffusion[END_REF], for initial data p 0 in a neighborhood of the target p. Related approximate and exact controllability issues have also been studied in Ainseba and Langlais [START_REF] Ainseba | On a population dynamics control problem with age dependence and spatial structure[END_REF], Ainseba and Iannelli [START_REF] Ainseba | Optimal screening in structured SIR epidemics[END_REF], Traore [START_REF] Traore | Null controllability of a nonlinear population dynamics problem[END_REF], Kavian and Traore [START_REF] Kavian | Approximate controllability by birth control for a nonlinear population dynamics model[END_REF]. The main novelty (with respect to the literature quoted above) brought in Theorem 1.1 is in our case u can be chosen such that (1.3) holds for every a ∈ (0, a + ) instead of (δ, a + ), with δ > 0, as it has been done, for instance, in [START_REF] Ainseba | Exact and approximate controllability of the age and space population dynamics structured model[END_REF]. We can thus find a control driving to zero the whole population, without excluding very young individuals. Moreover, we do not assume that the birth rate vanishes for small ages.

The remaining part of this work is organized as follows. In Section 2 we recall some basic results on the Lotka-McKendrick semigroup, with or without spatial diffusion, and we state a null controllability result associated to system (1.1) without spatial diffusion. Section 3 is devoted to study the null controllability of low frequencies for the solution of system (1.1). We prove the main result in Section 4, by using a version of the Lebeau-Robbiano strategy.

Notation:

In all what follows, C will denote a generic constant, depending only of the coefficients in (1.1), on Ω and ω, whose value may change from line to line.

Some background on the Lotka-McKendrick semigroup

In this section we recall, with no claim of originality, some existing results on the population semigroup for the linear age-structured model without and with spatial diffusion. In particular, we recall the structure of the spectrum of the semigroups generators, together with some controllability results concerning the free diffusion case.

The free diffusion case

In this paragraph we remind some results on the diffusion free case, which is described by the so-called McKendrick-Von Foster model. With one exception, we do not give proofs and we refer, for instance, to Song et al. [START_REF] Song | Spectral properties of population operator and asymptotic behaviour of population semigroup[END_REF] or Inaba [START_REF] Inaba | Age-structured population dynamics in demography and epidemiology[END_REF] for a detailed presentation of these issues.

The considered system is:

         ∂ t p(t, a) + ∂ a p(t, a) = -µ(a)p(t, a), t > 0, a ∈ (0, a † ), p(t, 0) = a † 0 β(a)p(t, a) da, t > 0, p(0, a) = p 0 (a), a ∈ (0, a † ), (2.1)
where µ and β satisfy the assumptions in Theorem 1.1.

The above system is described by the operator A 0 defined by

D(A 0 ) = ϕ ∈ L 2 [0, a † ] | ϕ(0) = a † 0 β(a)ϕ(a) da; - dϕ da -µϕ ∈ L 2 [0, a † ] , A 0 ϕ = - dϕ da -µϕ (ϕ ∈ D(A 0 )). (2.2)
Theorem 2.1. The operator A 0 defined by (2.2) has compact resolvent and its spectrum is constituted of a countable (infinite) set of isolated eigenvalues with finite algebraic multiplicity. The eigenvalues (λ 0 n ) n 1 of A 0 (counted without multiplicity) are the solutions of the characteristic equation

F (λ) := a † 0 β(a)e -λa π(a) da = 1.
(2.

3)

The eigenvalues (λ 0 n ) n 1 are of geometric multiplicity one, the eigenspace associated to λ 0 n being the one-dimensional subspace of L 2 (0, a † ) generated by the function

ϕ 0 n (a) = e -λ 0 n π(a) = e -λ 0 n a-a 0 µ(s) ds .
Finally, every vertical strip of the complex plane

α 1 Re(z) α 2 , α 1 , α 2 ∈ R, contains a finite number of eigenvalues of A 0 .
Theorem 2.2. The operator A 0 defined by (2.2) has a unique real eigenvalue λ 0 1 . Moreover, we have the following properties :

1. λ 0 1 is of algebraic multiplicity one; 2. λ 0 1 > 0 (resp. λ 0 1 < 0) if and only if F (0) > 1 (resp. F (0) < 1); 3. λ 0
1 is a real dominant eigenvalue:

λ 0 1 > Re(λ 0 n ), ∀n 2. (2.4)
It is well known (see, for instance, Song et al. [START_REF] Song | Spectral properties of population operator and asymptotic behaviour of population semigroup[END_REF] or Kappel and Zhang [START_REF] Kappel | Approximation of linear age-structured population models using Legendre polynomials[END_REF]) that A generates a C 0 semigroup of linear operators in L 2 [0, a † ] which we denote by T A0 = (T A0 t ) t 0 . We also have the following useful result (see, for instance, [10, p 23]):

Proposition 2.3. The semigroup T A0 generated on L 2 ([0, a † ]) by A 0 is compact for t a † .
According to Zabczyk [21, Section 2]), this implies in particular that

w a (A 0 ) = w 0 (A 0 ),
where ω a (A 0 ) := lim t→+∞ t -1 ln T A0 t L 2 (0,a † ) denotes the growth bound of the semigroup T A0 t and ω 0 (A 0 ) := sup{Reλ | λ ∈ σ(A 0 )} the spectral bound of A 0 . It is worth noticing that the above condition ensures that the exponential stability of T A0 is equivalent to the condition ω 0 (A 0 ) < 0. According to Theorem 2.1 and 2.2, it follows that the exponential stability of T A0 is equivalent to the condition λ 0 1 < 0, where λ 0 1 < 0 is the unique real solution to the characteristic equation defined by (2.3).

The free diffusion control problem associated to system (1.1) writes as

         ∂ t p(t, a) + ∂ a p(t, a) + µ(a)p(t, a) = v(t, a), t > 0, a ∈ (0, a † ), p(t, 0) = a † 0 β(a)p(t, a) da, t > 0, p(0, a) = p 0 (a), a ∈ (0, a † ), (2.5)
where v is the control function. Let us state a null controllability result for system (2.5):

Proposition 2.4. Under the assumptions of Theorem 1.1, for every τ > 0, there exists v ∈ L ∞ ((0, τ ); L 2 (0, a † )) such that the solution p of (2.5) satisfies p(τ, a) = 0 (a ∈ (0, a † ) a.e.).

Moreover, we have

v L ∞ ((0,τ );L 2 (0,a † )) sup σ∈[0,τ ] T A0 σ L 2 (0,a † ) τ p 0 L 2 (0,a † ) . (2.6)
Proof. Let τ > 0. For almost every (t, a) ∈ (0, τ ) × (0, a † ), we set

v(t, a) = - 1 τ (T A0 t p 0 )(a) (t ∈ (0, τ ), a ∈ (0, a † )). (2.7)
It is easy to check that the (mild) solution p of (2.5), given by

p(t, a) = T A0 t p 0 (a) + t 0 T A0 t-σ v(σ)(a) dσ, (2.8)
with v defined in (2.7), satisfies p(τ, •) = 0. Moreover, the cost stated in inequality (2.6) follows from the definition of the control v given by (2.7).

The population dynamics with diffusion

The existence of a semigroup on L 2 ((0, a † ) × Ω) describing the linear age-structured population model with diffusion coefficient and age dependent birth and death rates, with homogeneous Neumann boundary conditions has been proved in Huyer [START_REF] Huyer | Semigroup formulation and approximation of a linear age-dependent population problem with spatial diffusion[END_REF] (see also [START_REF] Guo | On the semigroup for age dependent population dynamics with spatial diffusion[END_REF] for the case of homogeneous Dirichlet boundary conditions).

More precisely, let X := L 2 ((0, a † ) × Ω) and define the following unbounded operator A on X:

D(A) = {ϕ ∈ X | ϕ ∈ C([0, a † ]; L 2 (Ω)) ∩ L 2 ([0, a † ], H 1 (Ω)), ϕ(0, x) = a † 0 β(a)ϕ(a, x) da; - dϕ da -µϕ + ∆ϕ ∈ X}, Aϕ = - dϕ da -µϕ + ∆ϕ (ϕ ∈ D(A)).
(2.9)

The generator A of the population semigroup can be seen as the sum of a population operator without diffusion -d/da -µI and a spatial diffusion term ∆. It turns out that spectral properties of A can be easily obtained from those of these two operators. Theorem 2.5. Let 0 = λ 0 < λ 1 λ 2 . . . be the increasing sequence of eigenvalues of -∆ with Neumann boundary conditions and let (ϕ n ) n 0 be a corresponding orthonormal basis of L 2 (Ω). Let (λ 0 n ) n 1 and (ϕ 0 n ) n 1 be respectively the sequence of eigenvalues and eigenfunctions of the free diffusion operator A 0 defined by (2.2) (see Theorem 2.1). Then the following assertions hold:

1. The eigenvalues of A are given by

σ(A) = {λ 0 i -λ j | i ∈ N * , j ∈ N}.
2. A has a dominant eigenvalue:

λ 1 = λ 0 1 > Re(λ), ∀λ ∈ σ(A), λ = λ 1 .
3. The eigenspace associated to an eigenvalue λ of A is given by

Span{ϕ 0 i (a)ϕ j (x) = e -λ 0 i a π(a)ϕ j (x) | λ 0 i -λ j = λ}. • • • • • • λ 0 λ 1 λ j → +∞ λ 0 1 λ 0 2 λ 0 3 Figure 1:
The spectrum of the free diffusion operator A 0 (green crosses) and of -∆ (red circles)

Since the operator A generates a C 0 semigroup of linear operators in X which we denote by T A = (T A t ) t 0 , this allows to define the concept of (mild) solution of (1.1) in the following standard way: we say that p is a mild solution of (1.1) if

p(t, •) = T A t p 0 + Φ t u (t 0, u ∈ L 2 ([0, ∞); X)), (2.10)
where the control operator B ∈ L(X) is defined by

Bu = χ ω u (u ∈ X),
and where

Φ t u = t 0 T A t-σ Bu(σ) dσ (t 0, u ∈ L 2 ([0, ∞); X)). (2.11)
It is worth noticing, for instance by using a spectral decomposition, that the semigroup T A is exponentially stable if λ 0 1 < 0, where we recall that λ 0 1 denotes the unique real solution to the characteristic equation defined by (2.3).

Remark 2.6. In order to prove the null controllability of system (1.1), we may assume, without loss of generality, that the so called reproductive number satisfies

a † 0 β(a)π(a) da < 1, (2.12)
which implies that both semigroups T A0 and T A are exponentially stable -see the above results.

Indeed, in the case where a † 0 β(a)π(a) da 1, we may consider the auxiliary system

                     ∂ t z(t, a, x) + ∂ a z(t, a, x) + μ(a)z(t, a, x) -∆z(t, a, x) = χ ω (x)v(t, a, x), t > 0, a ∈ (0, a † ), x ∈ Ω, ∂z ∂ν (t, a, x) = 0, t > 0, a ∈ (0, a † ), x ∈ ∂Ω, z(t, 0, x) = a † 0 β(a)z(t, a, x) da, t > 0, x ∈ Ω, z(0, a, x) = p 0 (a, x), a ∈ (0, a † ), x ∈ Ω, (2.13)
with μ(a) := µ(a) + λ, where λ 0 is large enough to have

a † 0 β(a)e -a 0 μ(s) ds da < 1.
Suppose that the above system (2.13) is null controllable with control function v. Then, system (1.1) is null controllable with control function u = e λt v, which has the same regularity as v.

From now on, without loss of generality (see Remark 2.6), we assume that both semigroups T A0 and T A are exponentially stable, which implies that there exists a constant C 0 and a constant M 0 such that for every t 0, we have

T A0 t L(L 2 [0,a † ]) C and T A t L(X)
M (t 0).

(2.14)

Low frequency control

In this section, we prove that the projection of the state trajectory of (1.1) on an infinite subspace of X (defined using the eigenfunctions of the Neumann Laplacian) can be steered to zero in any time and we estimate the norm of the associated control. More precisely, let {ϕ j } j 0 be an orthonormal basis in L 2 (Ω) formed of eigenvectors of the Neumann Laplacian and let (λ j ) j 0 be the corresponding non decreasing sequence of eigenvalues. In other words (ϕ j ) j 0 is an orthonormal basis in L 2 (Ω) such that for every j 0 we have

   -∆ϕ j = λ j ϕ j in Ω, ∂ϕ j ∂ν = 0 on ∂Ω. (3.1)
In the sequel, for any µ > 0, we denote by

N (µ) := Card{k : λ k µ}, (3.2)
E µ = Span{φ k : λ k µ},
and Π Eµ : L 2 (Ω) → L 2 (Ω) the orthogonal projection from L 2 (Ω) onto E µ . The main result of this section is: Proposition 3.1. Let µ > 0, and let T > 0. There exists u µ ∈ L ∞ ((0, T ); X) such that the solution p of (1.1) satisfies Π Eµ p(T, a, •) = 0 (a ∈ (0, a † ) a.e.)

Moreover, we have the following estimate:

u µ L ∞ ((0,T );X) Ce C √ µ T p 0 X .
The main ingredient of the proof is an inequality involving the eigenfunctions of the Neumann Laplacian. This inequality, obtained in Jerison and Lebeau [START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF], can also obtained by combining results and methods from Lebeau and Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]. Theorem 3.2. For any non-empty open subset ω of Ω, there exists C > 0 such that for any µ > 0, for any sequence (a j ) j 0 ⊂ R, we have

j:λj µ |a j | 2 Ce C √ µ ω j:λj µ a j ϕ j (x) 2 dx. (3.3)
We will also use the following classical lemma, whose proof can be found, for instance, in [19] -Section 2.

             ∂ t p j (t, a) + ∂ a p j (t, a) + (µ(a) + λ j )p j (t, a) = ω u(t, a, x)ϕ j (x) dx, t > 0, a ∈ (0, a † ), p j (t, 0) = a † 0 β(a)p j (t, a) da, t > 0, p j (0, a) = p j 0 (a), a ∈ (0, a † ), (3.4)
and where

p 0 (a, •) = +∞ j=0 p j 0 (a)ϕ j in L 2 (Ω), a.e. a ∈ (0, a † ).
The aim is to solve the following moment problem: find u ∈ L ∞ ((0, T ); X) such that for every j ∈ [0, N (µ)], we have ω u µ (t, a, x)ϕ j (x) dx = v j (t, a) ((t, a) ∈ (0, T ) × (0, a † ) a.e.), (3.5) where v j denotes a null control associated to the system

         ∂ t p j (t, a) + ∂ a p j (t, a) + (µ(a) + λ j )p j (t, a) = v j (t, a), t > 0, a ∈ (0, a † ), p j (t, 0) = a † 0 β(a)p j (t, a) da, t > 0, p j (0, a) = p j 0 (a), a ∈ (0, a † ). (3.6)
Recall from Proposition 2.4 and assumptions stated in Remark 2.6 that for every j 0, there exists v j ∈ L ∞ ((0, T ); L 2 (0, a † )) so that the solution p j of the above system (3.6) satisfies p j (T, a) = 0 for almost every a ∈ (0, a † ), with v j L ∞ ((0,T );L 2 (0,a † ))

C T p j 0 L 2 (0,a † ) , where C is a constant independent of λ j (since we can choose v j (t, a) = -e -tλ j T (T A0 t p j 0 )(a)). Let µ > 0. Define the map G : L 2 (ω) → R N (µ)+1 by Gu := ω u(x)ϕ j (x) dx 0 j N (µ)
.

It is easy to check that for every w = (w j ) 0 j N (µ) ∈ R N (µ)+1 , we have

G w = N (µ) j=0 w j ϕ j . (3.7) Inequality (3.3)
ensures that for every w ∈ R N (µ)+1 , we have

w R N (µ)+1 Ce C √ µ G w L 2 (ω) . (3.8)
By Lemma 3.3 with

W = Z = R N (µ)+1 , Y = L 2 (ω) and F = Id L(R N (µ)+1 ) , it follows that there exists H ∈ L(R N (µ)+1 , L 2 (ω)) such that GH = Id L(R N (µ)+1 ) . Moreover, H L(R N (µ) +1)
Ce C √ µ so that for every w ∈ R N (µ)+1 , there exists u := H(w) ∈ L 2 (ω) such that Gu = w and u L 2 (ω)

Ce C √ µ w R N (µ)+1 .
Let (t, a) ∈ (0, T ) × (0, a † ). Setting w := (v j (t, a)) 0 j N (µ) where v j is the null control defined by Proposition 2.4 (with µ(a) replaced by (µ(a) + λ j )), it follows that there exists u µ (t, a) ∈ L 2 (ω) such that Gu µ (t, a) = w, i.e.

ω u µ (t, a, x)ϕ j (x) dx = v j (t, a) (j ∈ [|0, N (µ)|], (t, a) ∈ (0, T ) × (0, a † ) a.e.), (3.9)
with

u µ (t, a, •) L 2 (ω) Ce C √ µ w R N (µ)+1 . (3.10)
From the above inequality (3.9), it follows that

u µ 2 L ∞ ((0,T );X) Ce C √ µ N (µ) j=0 v j 2 L ∞ ((0,T );L 2 (0,a † )) , (3.11)
where

v j 2 L ∞ ((0,T );L 2 (0,a † )) C T 2 p j 0 2
L 2 (0,a † ) by Proposition 2.4, so that from the above inequality we have

u µ 2 L ∞ ((0,T );X) Ce C √ µ T 2 (N (µ) + 1) p 0 2 X .
By Weyl's formula (see, for instance, Netrusov and Safarov [START_REF] Netrusov | Weyl asymptotic formula for the Laplacian on domains with rough boundaries[END_REF] for a reminder), there exists a constant K > 0 such that N (µ) Kµ n 2 , so that from the above formula we have

u µ L ∞ ((0,T );X) Ce C √ µ T p 0 X ,
for some constant C 0.

Proof of the main result

In this section we prove Theorem 1.1 by applying a version of the Lebeau-Robbiano strategy, initially proposed for the heat equation. A distinctive feature of the version of this methodology we propose here is that the projected systems are infinite dimensional, being described by equations similar to the Lotka-McKendrick system without spatial diffusion. To apply this strategy, we need these systems to be null controllable in arbitrarily small time so that our method is limited to the case of a control which is active for all ages.

Recall from Proposition 3.1 that, given a time T > 0 and a frequency µ > 0, there exists u µ ∈ L ∞ ((0, T ); X) such that the solution p of (1.1) belongs to the orthogonal of E µ at time T , for every a ∈ (0, a † ). The control cost behaves like e C √ µ , and can be compensated by the natural dissipation of the solution stated in the following proposition. Proposition 4.1. Let µ > 0. Suppose that Π Eµ p 0 (a, •) = 0 for almost every a ∈ (0, a † ). Then there exists a constant M 0 such that for every t > 0, the solution p of (1.1) with u ≡ 0 satisfies p(t, •, •) X M e -µt p 0 X (t 0).

Proof. Suppose that Π Eµ p 0 (a) = 0 for almost every a ∈ (0, a † ). With u ≡ 0, the solution p of (1.1) satisfies p(t, a, •) = j: λj >µ p j (t, a)ϕ j in L 2 (Ω), a.e. in (0, τ ) × (0, a † ),

where

         ∂ t p j (t, a) + ∂ a p j (t, a) + (µ(a) + λ j )p j (t, a) = 0 t > 0, a ∈ (0, a † ), p j (t, 0) = a † 0 β(a)p j (t, a) da, t > 0,
p j (0, a) = p j 0 (a), a ∈ (0, a † ).

(

Let λ j > µ. It is easy to check that the solution p j of (4.1) satisfies p j (t, a) = e -tλj T A0 t (p j 0 )(a), so that using (2.14) we have

p j (t, •) L 2 (0,a † ) M e -λj t p j 0 L 2 (0,a † ) M e -µt p j 0 L 2 (0,a † ) , (4.2
) since λ j > µ. Using p(t, a, •) 2 L 2 (Ω) = j: λj >µ |p j (t, a)| 2 ,
it follows that we have

p(t, •, •) 2 L 2 ((0,a † )×Ω) = j: λj >µ p j (t, •) 2 L 2 (0,a † ) M 2 e -2µt j: λj >µ p j 0 2 L 2 (0,a † ) M 2 e -2µt p 0 2 X , (4.3)
so that the estimation of Proposition 4.1 holds.

The following corollary is the key point of the strategy of Lebeau and Robbiano, originally developed for the null controllability of the heat equation [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. Given a time T > 0, we construct a control in two steps in the time interval (0, T ): we control the first frequencies in the time interval (0, T /2) with the cost obtained in Proposition 3.1, and we let the system to evolve freely in the time interval (T /2, T ) in order to dissipate the energy possibly transferred to the high frequencies (see Proposition 4.1).

Proof of theorem (1.1). Let τ > 0. Consider the sequences T j = τ 2 j and µ j = β(2 j ) 2 (j 1), where β > 0 is determined later. Denote by τ 0 = 0 and τ j = j k=1

T k , for every j 1, so that we have (0, τ ) = j 0 (τ j , τ j+1 ). Following the strategy of Lebeau and Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], we will define a control by induction on each interval (τ j , τ j+1 ) which drives the initial state to zero in time τ .

Firstly, during the time interval (0, τ 1 ) = (0, T 1 ), we apply a control u 1 as given by Corollary 4.2 with µ = µ 1 , so that we have

u 1 L ∞ ((0,τ1);X) Ce C √ µ1 T 1 p 0 X , and
p(τ 1 , •, •) X Ce C √ µ1-T 1 µ 1 2 p 0 X .
Given j 1, during the time interval (τ j-1 , τ j ), we apply by induction a control function denoted by u j as given by Corollary 4.2 with µ = µ j so that we have by construction of the sequences (T j ) j 1 and (µ j ) j 1 . Then we choose β > 0 large enough so that β := β 2 τ -C β > 0, (4.17) so that from (4.15), (4.16) and (4.17) we have p(τ j , •, •) X KC j e -β2 j+1 p 0 X , (4.18)

u j L ∞ ((τj-1,τj );X) Ce C √ µj T j p(τ j-1 , •, •) X , (4.13)
for some constant K 0.

Going back to (4.13) and using the estimate given by (4.18), it follows that we have thanks to estimation (4.18). This implies that p(τ, •, •) = 0, since τ j → τ as j → +∞. for some constant K 0. The same type of estimation is shown by Seidman [START_REF] Seidman | How violent are fast controls. III[END_REF] for the null controllability of the heat equation, using also an adaptated Lebeau-Robbiano strategy.

Lemma 3 . 3 .

 33 Suppose that W, Y, Z are Hilbert spaces, F ∈ L(W, Z) and that G ∈ L(Y, Z). Denote by A * the adjoint of an operator A. Then the following statements are equivalent: (i) there exists c > 0 such that F * z W c G * z Y (z ∈ Z); (ii) there exists H ∈ L(W, Y ) such that GH = F and H L(W,Y ) c. Proof of Proposition 3.1. Note that the solution p of (1.1) writes p(t, a, •) = +∞ j=0 p j (t, a)ϕ j in L 2 (Ω), a.e. in (0, T ) × (0, a † ), where

2 p(τ j- 1 ,√ µ k -1 2 jk=1 T k µ k p 0 T

 2120 and p(τ j , •, •) X Ce C √ µj -T j µ j •, •) X .(4.14)From (4.14), it follows that for every j 1 we havep(T j , •, •) X C j e C j k=1 k µ k = 2(2 j -1)(C β -

1 u

 1 C j-1 e -2 β(2 j-1 -1) p 0 X = K τ C j 2 -j e 2 j (C √ β-β) p 0 X ,(4.19)with K = Ke 2 β . We choose β > 0 large enough such thatβ := β -C β = β 2 τ -2C β > 0,(4.20)so that from (4.[START_REF] Tenenbaum | On the null-controllability of diffusion equations[END_REF]) and (4.20) we haveu j L ∞ ((τj-1,τj );X) K √ τ C j 2 -j e -β2 j p 0 X . (4.21)The above estimate (4.21) ensures that sup j j L ∞ ((τj-1,τj );X) < +∞, (4.22) so that defining the control u byu = +∞ j=1 u j 1 (τj-1,τj )(4.23)gives a control function in L ∞ ((0, τ ); X). The corresponding trajectory p is continuous in time with values in X and satisfies p(τ j , •, •) X -→

Remark 4 . 3 . 2 √ 2 )

 4322 Given τ > 0, in the above proof, we can choose β = 2( 4C τ ) 2 so that the condition (4.20) :β := β 2 τ -2C √ β > 0 is fulfilled (in this case, we have β = 4c 2 τ (4 -).With this choise of β, it follows from (4.21) that the control u defined by (4.23) satisfiesu L ∞ ((0,τ );X) Ke K τ p 0 X ,(4.25)

Acknowledgements.The authors are grateful to Sebastian Anit ¸a for numerous discussions and comments which strongly helped them to simplify the proof of the main results.

Corollary 4.2. For every µ > 0 and every T ∈ (0, τ ), there exists u µ ∈ L ∞ ((0, T ); X) such that u µ L ∞ ((0,T );X)

Ce C √ µ T p 0 X ,

where p is the solution of (1.1) with control function u µ .

Proof. Let T ∈ (0, τ ). We first use Proposition 3.1 on the time interval (0, T /2). This gives us a control function v µ ∈ L ∞ ((0, T /2); X) such that the solution p of (1.1) with control function

and

In the second half interval (T /2, T ), we choose a null control in order to take advantage of the natural dissipation given by Proposition 4.1. More precisely, for almost every t ∈ (0, T ), we set

Thanks to (4.6) and (4.7), it is clear that we have

and since u µ = 0 on the time interval (T /2, T ), formula (2.10) gives p(T ) = T A T /2 p(T /2), (4.9) so that using Proposition 4.1 together with (4.5) and (4.9), it follows that we have

Moreover, it follows from formula (2.10) and (4.7) that we have

and it follows from the assumption (2.14) and Cauchy-Schwarz inequality that we have

so that using (4.6), (4.10) and (4.11) we have

for some constant C 0.

We then use a time-splitting procedure, as described in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], to get the null controllability of system (1.1).