Piermarco Cannarsa
email: cannarsa@mat.uniroma2.it

AND Patrick Martinez
email: patrick.martinez@math.univ-toulouse.fr

Judith Vancostenoble
email: judith.vancostenoble@math.univ-toulouse.fr

PRECISE ESTIMATES FOR BIORTHOGONAL FAMILIES UNDER ASYMPTOTIC GAP CONDITIONS

Keywords: Subject Classification. 11B05, 30B10, 30D15 Biorthogonal families, gap conditions

published or not. The documents may come

Biorthogonal families are a classical tool in analysis. In particular, they play a crucial role in the so-called moment method, which was developed by Fattorini-Russell [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] to study controllability for parabolic equations.

Given any sequence of nonnegative real numbers, (λ n ) n≥1 , we recall that a sequence (σ m ) m≥1 is biorthogonal to the sequence (e λnt ) n≥1 in L 2 (0, T ) if

∀m, n ≥ 1, T 0 σ m (t)e λnt dt = 1 if m = n 0 if m = n .
The goal of this paper is to provide explicit and precise upper and lower bounds for the biorthogonal family (σ m ) m≥1 under the following gap conditions:

• a 'global gap condition':

(1. 1) ∀n ≥ 1, 0 < γ min ≤ λ n+1 -λ n ≤ γ max ,

• and an 'asymptotic gap condition':

(1. 2) ∀n ≥ N * , γ * min ≤ λ n+1 -λ n ≤ γ * max ,
where γ * max -γ * min < γ max -γ min . Before explaining why we are interested in such a question, let us describe some of the main results of the literature on this subject.

The context.

Among the most important applications of biorthogonal families to control theory are those to the null controllability and sensitivity of control costs to parameters. Major contributions in such directions are the following:

• Fattorini-Russell [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF], Hansen [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF], and Ammar Khodja-Benabdallah-González Burgos-de Teresa [START_REF] Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF] studied the existence of biorthogonal sequences and their application to controllability for various equations; • for nondegenerate parabolic equations and dispersive equations, Seidman [START_REF] Seidman | Two results on exact boundary control of parabolic equations[END_REF], Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF], Seidman-Avdonin-Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], Miller [START_REF] Miller | Geometric bounds on the growth rate of null controllability cost for the heat equation in small time[END_REF], Tenenbaum-Tucsnak [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF], and Lissy [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transportdiffusion equation[END_REF] studied the dependence of the null controllability cost C T with respect to the time T (as T → 0, the so-called 'fast control problem') and with respect to the domain, obtaining extremely sharp estimates of the constants c(Ω) and C(Ω) that appear in e c(Ω)/T ≤ C T ≤ e C(Ω)/T ;

• Coron-Guerrero [START_REF] Coron | Singular optimal control: A linear 1-D parabolic-hyperbolic example[END_REF], Glass [START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit[END_REF], Lissy [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transportdiffusion equation[END_REF] investigated the vanishing viscosity problem:

y t + M y x -εy xx = 0, x ∈ (0, L), y(0, t) = f (t),
obtaining sharp estimates of the null controllability cost with respect to the time T , the transport coefficient M , the size of the domain L, and the diffusion coefficient ε; • in [START_REF] Cannarsa | Global Carleman estimates for degenerate parabolic operators with applications[END_REF][START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF], we studied the dependence of the controllability cost with respect to the degeneracy parameter α for the degenerate parabolic equation u t -(x α u x ) x = 0, x ∈ (0, );

• and recently Gueye-Lissy [START_REF] Gueye | Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation[END_REF] studied a 1-D parabolic-hyperbolic degenerate equation.

There is a common feature in these works: they depend on some parameter p, and this parameter forces the eigenvalues to satisfy (1. 1) (sometimes after normalization) with gap bounds γ min (p) and γ max (p) such that γ min (p) → 0 and/or γ max (p) → ∞.

This fact makes it necessary to have general and precise estimates with respect to the main parameters that appear in the problem.

In [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF], we proved the following general result: given T > 0 and a family (λ n ) n≥1 of nonnegative real numbers that satisfy the 'global gap condition' (1. 1), then:

• every family (σ m ) m≥1 , biorthogonal to (e λnt ) n≥1 in L 2 (0, T ), satisfies the lower estimate

(1. 3) σ m 2 L 2 (0,T ) ≥ b m e -2λmT e 1 2T (γmax) 2 ,
with an explicit value of b m = b m (T, γ max , m) (rational in T ); • there exists a family (σ m ) m≥1 , biorthogonal to (e λnt ) n≥1 in L 2 (0, T ), that satisfies the upper estimate This generates several comments:

• concerning the results: √ λ n are not uniformly bounded from below by a positive constant); * and, on the contrary, (1. 5) forces the sequence ( √ λn n ) n to converge (to √ r), while one can construct sequences satisfying (1. 1) for which ( √ λn n ) n does not converge, considering, e.g., the recurrence formula λ n+1 = √ λ n +ε n with ε n ∈ {1, 2}, and choosing ε n to be constant on sufficiently large intervals, in such a way that 1 and 2 are limits of subsequences of ( √ λn n ) n .)

1.3. Motivations and main results of this paper.

Even though the aforementioned results give a fairly good picture of the properties of the family (σ m ) m , some delicate issues remain to be analyzed and will be addressed in this paper. For instance, one would like to understand the dependence of the family (σ m ) m with respect to relevant parameters that come into play. Typical examples of such problems are the following ones.

• For the 1D degenerate parabolic equation

u t -(x α u x ) x = 0, x ∈ (0, ),
the eigenvalues λ α,n of the associated elliptic operator (with suitable boundary conditions) can be expressed using the zeros of Bessel functions ( [START_REF] Gueye | Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations[END_REF]) and depend on the degeneracy parameter α ∈ (0, 2). One can then prove (see [START_REF] Cannarsa | Global Carleman estimates for degenerate parabolic operators with applications[END_REF][START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF]) that the global gap condition (1. 1) is satisfied only with

γ max (α) ≥ c(2 -α) 2/3 ,
with c > 0, while the asymptotic gap condition (1. 2) is satisfied with

γ * max (α) ≤ c * (2 -α),
where c * > 0, after the rank

N * (α) = 1 2 -α ; in this case γ max (α) γ * max (α) -→ +∞ as α → 2 -;
hence it is natural to think that the better asymptotic gap (1. 2) could be used to improve the estimate (1. 3) of the associated biorthogonal sequences, but the fact that

N * (α) -→ +∞ as α → 2 -
is certainly to be taken into account.

• In 2D problems such as the Grushin equation (see [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF][START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF]), where the solution is decomposed into Fourier modes, one has to give uniform bounds for a certain sequence of elliptic problems, the eigenvalues of which satisfy (1. 1) and (1. 2) with some γ min (m), γ * min (m) and

N * (m) such that γ min (m) γ * min (m) -→ 0 as m → ∞ and N * (m) -→ +∞ as m → ∞;
once again, it is natural to think that the better asymptotic gap (1. 2) could be used to improve the estimate (1. 4) of the associated biorthogonal sequence, but the fact that N * (m) → +∞ as m → ∞ is certainly to be taken into account.

The above discussion motivates the general question whether estimates (1. 3) and (1. 4) can be improved when (1. 1) is combined with the asymptotic condition (1. 2). This is exactly what we prove in this paper: roughly speaking, (1. 3) and (1. 4) hold true replacing γ min by γ * min and γ max by γ * max . Moreover, the fact that the 'good' gap condition (1. 2) holds true only after the N * first eigenvalues has a cost, and we obtain a precise estimate for that cost. Our main results (Theorem 2.1 and 2.2) are the following: under (1. 2), we prove that:

• every family (σ m ) m≥1 , biorthogonal to (e λnt ) n≥1 in L 2 (0, T ), satisfies the lower estimate

(1. 6) σ m 2 L 2 (0,T ) ≥ b * m e -2λmT e 2 T (γ * max ) 2 , where the 'cost' b * m = b * m (T, γ max , γ * max , N * , m
) is a rational function of T that we determine explicitly, and • there exists a biorthogonal family that satisfies in (1. 6) and (1. 7) is quite natural and has already been pointed out by Seidman-Avdonin-Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], Tenenbaum-Tucsnak [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF], and Lissy [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transportdiffusion equation[END_REF] (see also Haraux [START_REF] Haraux | Séries lacunaires et contrôle semi-interne des vibrations dune plaque rectangulaire[END_REF] and Komornik [START_REF] Komornik | Fourier Series in Control Theory[END_REF] for a closely related context). On the other hand, the precise estimate of the behavior of b * m and B * m with respect to parameters, that we develop in this paper, is completely new and will be crucial for the sensitivity analysis of control costs to be performed in [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF].

Our proofs are based on complex analysis techniques and hilbertian methods developed by Seidman-Avdonin-Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF] and Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF]. We have also used an idea from Tenenbaum-Tucsnak [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF] and Lissy [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transportdiffusion equation[END_REF], based on the introduction of an extra parameter depending on T and the gap conditions.

Finally, as mentioned above, the gap conditions (1. 1)-(1. 2) are practical to obtain such estimates from above and below, but one could investigate more general conditions, of the form

(1. 8) α n n ≤ λ n+1 -λ n ≤ β n n,
with lim inf n α n > 0 and lim sup β n < ∞. This would include the case of assumptions (1. 1) and (1. 5) and could allow to study more general problems.

1.4. Plan of the paper.

The paper is organized as follows:

• in section 2, we state our results;

• section 3 is devoted to the proof of Theorem 2.1 (construction of a biorthogonal family and derivation of upper bounds); • section 4 is devoted to the proof of Theorem 2.2 (lower bounds for biorthogonal families).

Setting of the problem and main results

2.1. Existence of a suitable biorthogonal family and upper bounds.

We will prove the following result, that in some sense precises results of Fattorini and Russell [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] (in short time) and are in the spirit of results of Tenenbaum-Tucsnak [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF], Lissy [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transportdiffusion equation[END_REF] (with a slightly weakened assumption on the eigenvalues).

Theorem 2.1. Assume that ∀n ≥ 1, λ n ≥ 0, and that there is some 0 < γ min < γ * min such that

(2. 1) ∀n ≥ 1, λ n+1 -λ n ≥ γ min , and
(2. 2) ∀n ≥ N * , λ n+1 -λ n ≥ γ * min . Denote (2. 3) M * := (1 - γ min γ * min )(N * -1).
Then there exists a family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ):

(2. 4) ∀m, n ≥ 1, T 0 σ + m (t)e λnt dt = δ mn .
Moreover, it satisfies: there is some universal constant C independent of T , γ min , γ * min , N * and m such that, for all m ≥ 1, we have

(2. 5) σ + m 2 L 2 (0,T ) ≤ e -2λmT e C T (γ * min ) 2 e C √ λm γ * min B * (T, γ min , γ * min , N * , m), where (2. 6) B * (T, γ min , γ * min , N * , m) =      C u (8M * )! (λm(γ * min ) 2 T 2 ) 4M * + 1 e CuM * e Cu λ N * γ min √ λm ( 1 T 3/2 + 1 (γ * min ) 2 T 2 ) if T ≤ 1 (γ * min ) 2 C u ( (γ * min ) 8M * (8M * )! λ 4M * m ) + 1 e CuM * e Cu λ N * γ min √ λm ((γ * min ) 2 + (γ * min ) 3 ) if T ≥ 1 (γ * min ) 2 .
Remark 2.1. Theorem 2.1 completes and improves several earlier results, in particular Theorem 1.5 of Fattorini-Russell [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] and [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] expliciting the dependence of the L 2 bound with respect to γ min , γ * min and in short time. It is useful in several problems, in which γ min → 0 with respect to some parameter, which occurs is several cases, see, e.g. [START_REF] Fattorini | Boundary control of temperature distributions in a parallelepipedon[END_REF], [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF]. We will use the construction that was used by Seidman, Avdonin and Ivanov in [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], which has the advantage to be completely explicit (which is not the case for the construction of [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF][START_REF] Fattorini | Boundary control of temperature distributions in a parallelepipedon[END_REF][START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF][START_REF] Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF], since there is a contradiction argument), combined with some ideas coming from the construction of Tenenbaum-Tucsnak [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF] and Lissy [START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF], adding some parameter, in order to obtain precise results.

General lower bounds.

We generalize a result of Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] to prove the following Theorem 2.2. Assume that

∀n ≥ 1, λ n ≥ 0,
and that there are

0 < γ min ≤ γ * max ≤ γ max such that (2. 7) ∀n ≥ 1, γ min ≤ λ n+1 -λ n ≤ γ max , and
(2. 8) ∀n ≥ N * , λ n+1 -λ n ≤ γ * max .
Then any family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ) (hence that satisfies (2. 4)) satisfies:

(2. 9) σ + m 2 L 2 (0,T ) ≥ e -2λmT e 2 T (γ * max ) 2 b * (T, γ max , γ * max , N * , λ 1 , m) 2
, where b * is rational in T (and explictly given in the key Lemma 4.4).

Remark 2.2. Theorem 2.2 completes a result of Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] and is useful in several problems, in which γ max → ∞ with respect to some parameter, which occurs is several cases, see, e.g. [START_REF] Fattorini | Boundary control of temperature distributions in a parallelepipedon[END_REF], and [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF]. It is to be noted that the behavior with respect to m can perhaps be improved, comparing with Theorem 1.1 of Hansen [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF]. It would be interesting to investigate this. It begins with the following remarks: if the family (σ + m ) m≥1 is biorthogonal to the family (e λnt ) n≥1 , then

∀m, n ≥ 1, T 0 σ + m (T -t)e λn(T -t) dt = δ mn , hence ∀m, n ≥ 1, T 0 σ + m (T -t)e λmT e -λnt dt = δ mn ,
hence the family (s m ) m≥0 defined by s m (t) := σ + m (T -t)e λmT is biorthogonal to the family (e -λnt ) n≥1 in L 2 (0, T ). Now extend s m by 0 outside (0, T ), and consider its Fourier transform

∀z ∈ C, F(s m )(z) := R s m (t)e -izt dt.
For all m ≥ 1, F(s m ) is the Fourier transform of a compactly supported function, hence it is an entire function over C, and it satisfies

∀m, n ≥ 1, F(s m )(-iλ n ) = δ mn ,
and it is of exponential type:

|F(s m )(z)| ≤ T 0 |s m (t)| dt e T |z| ;
and also

F(s m )(z) = T /2 -T /2 s m (τ + T 2 )e -iz(τ + T 2 ) dτ, hence F(s m )(-z)e -iz T 2 = T /2 -T /2 s m (τ + T 2 )e izτ dτ, and
|F(s m )(-z)e -iz T 2 | ≤ T /2 -T /2 |s m (τ + T 2 )| dτ e T 2 |z| .
Now we recall the Paley-Wiener theorem ( [START_REF] Young | An Introduction to Nonharmonic Fourier Series[END_REF]): if f : C → C is an entire function of exponential type, such that there exist nonnegative constants C, A such that

∀z ∈ C, |f (z)| ≤ Ce A|z| , and if f ∈ L 2 (R), then there exists φ ∈ L 2 (-A, A) such that f (z) = A -A φ(τ )e izτ dτ.
One of the objects of [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF] is to prove the existence of a sequence (f m ) m of entire functions satisfying (3. 1)

     ∀m, n ≥ 1, f m (-iλ n ) = δ mn , ∀z ∈ C, |f m (-z)e -iz T 2 | ≤ C m e T 2 |z| , ∀m ≥ 1, f m ∈ L 2 (R)
(see Theorem 2 and Lemma 3 in [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF]) under some general assumptions on the sequence (λ n ) n . If we can apply such a result in our context (hence with our sequence (λ n ) n ), then the two last properties together with the Paley-Wiener theorem will imply that there exists some

φ m ∈ L 2 (-T 2 , T 2 ) such that f m (-z)e -iz T 2 = T /2 -T /2 φ m (τ )e izτ dτ, hence f m (z) = T 0 φ m (t - T 2 )e -izt dt,
and then

T 0 φ m (t - T 2 )e -λnt dt = f m (-iλ n ) = δ mn , hence (φ m (t -T 2
)) m will be biorthogonal to the family (e -λnt ) n , and (σ + m (t)) m defined by

σ + m (t) = φ m ( T 2 -t)e -λmT
will be biorthogonal to the family (e λnt ) n in L 2 (0, T ), as desired. Moreover

σ + m 2 L 2 (0,T ) = e -2λmT T /2 -T /2 φ m (τ ) 2 dτ ≤ Ce -2λmT f m 2 L 2 (R)
using the Parseval theorem. Now, it remains to construct such entire functions f m . The idea is to consider the natural infinite product that satisfies the first condition of (3. 1), f m (-iλ n ) = δ mn , and to multiply it by a so-called 'mollifier', in such a way that the other two conditions of (3. 1) will be also satisfied. Hence one has to estimate the growth of the natural infinite product, and then to choose a choose a suitable mollifier. This is what is performed in [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF]. For our problem, our task will be to add the dependency into the parameters γ min , γ * min and T , and to understand specifically the behaviour of the natural infinite product, the mollifier and at the end of σ + m L 2 (0,T ) with respect to γ min and T . We will modify a little the construction of [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], in order to obtain optimal results in our context, see Lemma 3.4, and specifically the definition (3. 19) of the mollifier, where the additionnal parameter N will be chosen of the size

1 T (γ * min ) 2 , see (3. 30). 3.2. The counting function. Consider ∀ρ > 0, N n (ρ) := card {k, 0 < |λ n -λ k | ≤ ρ}.
We prove the following:

Lemma 3.1. a) Assume that the gap assumption (2. 1) is satisfied; then

(3. 2) ∀n ≥ 0, ∀ρ > 0, N n (ρ) ≤ 2 √ ρ γ min .
b) Assume that the gap assumptions (2. 1)-(2. 2) are satisfied; then • when n = N * :

(3. 3) ∀ρ > 0, N N * (ρ) ≤ √ ρ γmin + √ ρ γ * min if ρ ≤ λ N * N * -1 + √ ρ γ * min if ρ ≥ λ N * ,
• when n > N * :

(3. 4) ∀n > N * , ∀ρ > 0, N n (ρ) ≤        2 √ ρ γ * min if ρ ≤ λ n -λ N * √ ρ γmin + √ ρ γ * min if λ n -λ N * ≤ ρ ≤ λ n n -1 + √ ρ γ * min if ρ ≥ λ n , • when n < N * -when λ n ≤ λ N * -λ n , then (3. 5) ∀ρ > 0, N n (ρ) ≤        2 √ ρ γmin if ρ ≤ λ n n -1 + √ ρ γmin if λ n ≤ ρ ≤ λ N * -λ n N * -1 + √ ρ γ * min if ρ ≥ λ N * -λ n , -when λ n ≥ λ N * -λ n , then (3. 6) ∀ρ > 0, N n (ρ) ≤        2 √ ρ γmin if ρ ≤ λ N * -λ n N * -n + √ ρ γmin + √ ρ γ * min if λ N * -λ n ≤ ρ ≤ λ n N * -1 + √ ρ γ * min if ρ ≥ λ n . Remark 3.1.
λ k -λ n = λ k 2 -λ n 2 = ( λ k -λ n )( λ k + λ n ),
and the gap assumption (2. 1) insures that

λ k -λ n ≥ (k -n)γ min , λ k + λ n ≥ (k -n)γ min + 2 λ n , hence λ k -λ n ≥ (k -n) 2 γ 2 min , and
k > n and λ k -λ n ≤ ρ =⇒ k -n ≤ √ ρ γ min .
Similarly,

k < n and λ n -λ k ≤ ρ =⇒ n -k ≤ √ ρ γ min .
Hence

N n (ρ) ≤ 2 √ ρ γ min .
This proves (3. 2). Now we prove (3. 3)-(3. 6): let us introduce

N + n (ρ) := card {k > n, λ k -λ n ≤ ρ}, N - n (ρ) := card {k < n, λ n -λ k ≤ ρ}.
We distinguish the three cases.

• When n = N * : from the previous study, we see that

N + N * (ρ) ≤ √ ρ γ * min , and
N - N * (ρ) ≤ √ ρ γmin if ρ ≤ λ N * N * -1 if ρ ≥ λ N * ;
this gives that

N N * (ρ) ≤ √ ρ γmin + √ ρ γ * min if ρ ≤ λ N * N * -1 + √ ρ γ * min if ρ ≥ λ N * ,
which gives (3. 3). • When n > N * : now we have

N + n (ρ) ≤ √ ρ γ * min , and
N - n (ρ) ≤      √ ρ γ * min if ρ ≤ λ n -λ N * √ ρ γmin if λ n -λ N * ≤ ρ ≤ λ n n -1 if ρ ≥ λ n , which gives (3. 4). • When n < N * : now we have N - n (ρ) ≤ √ ρ γmin if ρ ≤ λ n n -1 if ρ ≥ λ n ,
and

N + n (ρ) ≤ √ ρ γmin if ρ ≤ λ N * -λ n N * -n + √ ρ γ * min if ρ ≥ λ N * -λ n , hence when λ n ≤ λ N * -λ n we have N n (ρ) ≤        2 √ ρ γmin if ρ ≤ λ n n -1 + √ ρ γmin if λ n ≤ ρ ≤ λ N * -λ n N * -1 + √ ρ γ * min if ρ ≥ λ N * -λ n ,
which gives (3. 5), and similar estimates when λ n ≥ λ N * -λ n , which give (3. 6).

Before going further, let us give another estimate of the counting function, which reveals to be more practical and more natural, since it gives a better understanding of the role of the different parameters: Lemma 3.2. Assume that the gap assumptions (2. 1)-(2. 2) are satisfied; then

• when n = N * :

(3. 7) ∀ρ > 0, N N * (ρ) ≤ √ ρ γmin + √ ρ γ * min if ρ ≤ λ N * (1 -γmin γ * min )(N * -1) + 2 √ ρ γ * min if ρ ≥ λ N * , • when n > N * : (3. 8) ∀ρ > 0, N n (ρ) ≤        2 √ ρ γ * min if ρ ≤ λ n -λ N * √ ρ γmin + √ ρ γ * min if λ n -λ N * ≤ ρ ≤ λ n (1 -γmin γ * min )(N * -1) + 2 √ ρ γ * min if ρ ≥ λ n , • when n < N * : (3. 9) ∀ρ > 0, N n (ρ) ≤ 2 √ ρ γmin if ρ ≤ max{λ n , λ N * -λ n } (1 -γmin γ * min )(N * -1) + (1 + √ 2) √ ρ γ * min if ρ ≥ max{λ n , λ N * -λ n } ,
and also

(3. 10) ∀ρ > 0, N n (ρ) ≤ 2 √ ρ γmin if ρ ≤ λ N * (1 -γmin γ * min )(N * -1) + 2 √ ρ γ * min if ρ ≥ λ N * . Remark 3.2. Lemma 3.2 enlightens the role of the quantity (1 -γmin γ * min )(N * -1) (denoted M * in (2.
3)); when γ min = γ * min or if N * = 1, this quantity is equal to zero, and we logically find estimates similar to the ones of Lemma 3.1 (i.e. the '1 gap condition'); in the more interesting case where γ min < γ * min and N * > 1, this quantity measures the increase of the counting function with respect to the '1 gap condition'.

Let us note also that we expect that (3. 9) holds true with 2 instead of 1 + √ 2, however we could not prove it in full generality.

Proof of Lemma 3.2.

• When n = N * , it is sufficient to note that

λ N * ≥ γ min (N * -1),
hence, when ρ ≥ λ N * , we have

N * -1 = (1 - γ min γ * min )(N * -1) + γ min γ * min (N * -1) ≤ (1 - γ min γ * min )(N * -1) + √ λ N * γ * min ≤ (1 - γ min γ * min )(N * -1) + √ ρ γ * min ;
this and (3. 3) imply (3. 7). • When n > N * : when ρ ≥ λ n , we have

λ n ≥ λ N * + (n -N * )γ * min ≥ (N * -1)γ min + (n -N * )γ * min , hence n -1 ≤ √ λ n γ * min + (1 - γ min γ * min )(N * -1) ≤ √ ρ γ * min + (1 - γ min γ * min )(N * -1);
this estimate and (3. 4) imply (3. 8).

• When n < N * , we obtain (3. 10) proceeding in the same way: when ρ ≤ λ N * , then clearly N n (ρ) is less than the number of terms that would be at both sides, for which the gap of their square root would be γ min , hence

N n (ρ) ≤ 2 √ ρ γ min ;
when ρ ≥ λ N * , then clearly one has all the N * -1 first terms, and the others, for which the gap of their square root is γ * min , hence

N n (ρ) ≤ N * -1 + √ ρ γ * min ; but then N * -1 = (1 - γ min γ * min )(N * -1) + γ min γ * min (N * -1) ≤ (1 - γ min γ * min )(N * -1) + √ λ N * γ * min ≤ (1 - γ min γ * min )(N * -1) + √ ρ γ * min ,
which gives (3. 10); • finally we prove (3. 9): in the same way, if ρ ≤ max{λ n , λ N * -λ n } one has immediately

N n (ρ) ≤ 2 √ ρ γ min ;
when ρ ≥ max{λ n , λ N * -λ n }, then we already know from (3. 5) and (3. 6) that

N n (ρ) ≤ N * -1 + √ ρ γ * min , hence N n (ρ) ≤ (1 - γ min γ * min )(N * -1) + γ min γ * min (N * -1) + √ ρ γ * min ; since λ N * ≥ γ min (N * -1) and ρ ≥ max{λ n , λ N * -λ n } =⇒ ρ ≥ 1 2 λ n + (λ N * -λ n ) = 1 2 λ N * ,
we deduce that

γ min (N * -1) ≤ λ N * ≤ 2ρ, hence N n (ρ) ≤ (1 - γ min γ * min )(N * -1) + (1 + √ 2) √ ρ γ * min ,
which is (3. 9). This concludes the proof of Lemma 3.2.

A Weierstrass product.

Motivated by [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], we consider

(3. 11) F m (z) := ∞ k=1,k =m 1 - iz -λ m λ k -λ m 2 .
Then the growth in k of λ k ensures that this infinite product converges uniformly over all the compact sets, hence F m is well-defined and entire over C. Moreover

F m (-iλ n ) = ∞ k=1,k =m 1 - λ n -λ m λ k -λ m 2 = 0 if m = n, 1 if m = n, , hence (3. 12) ∀m, n ≥ 1, F m (-iλ n ) = δ mn .
We are going to estimate the growth of F m . We prove the following

B m =          e ( 8 γ min + Cu γ * min ) √ λm+ 8 γ min √ λ N * -λm if m < N * e ( 4 γ min + 4+Cu γ * min ) √ λ N * if m = N * e ( 4 γ min + 8+Cu γ * min ) √ λm-4 γ min √ λm-λ N * if m > N * and (3. 18) q m (|z|) =            1 + 2 |z| 2 +λ 2 m max(λm,λ N * -λm) 2 M * if m < N * 1 + 2 |z| 2 +(λ N * ) 2 (λ N * ) 2 M * if m = N * 1 + 2 |z| 2 +λ 2 m λ 2 m M * if m > N *
, and this easily implies (3. 15) and (3. 16).

Proof of Lemma 3.3. Note that

F m (z -iλ m ) = ∞ k=1,k =m 1 + z 2 (λ k -λ m ) 2 ,
hence (following [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF])

ln |F m (z -iλ m )| = ∞ k=1,k =m ln 1 + z 2 (λ k -λ m ) 2 ≤ ∞ k=1,k =m ln 1 + |z| 2 (λ k -λ m ) 2 = ∞ 0 ln 1 + |z| 2 ρ 2 dN m (ρ) = 2 ∞ 0 N m (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ
Then we distinguish several cases:

• Under only (2. 1) we deduce from (3. 2) that

2 ∞ 0 N m (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ ≤ 4 γ min ∞ 0 1 √ ρ |z| 2 |z| 2 + ρ 2 dρ = 4 γ min ∞ 0 1 √ s 1 1 + s 2 ds |z|.
Then changing z -iλ m into z,

ln |F m (z)| ≤ 4 γ min ∞ 0 1 √ s 1 1 + s 2 ds ( |z| + λ m ),
which gives (3. 13).

• Under (2. 1)-(2. 2) and when m = N * , we derive from (3. 7) that

2 ∞ 0 N N * (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ = 2 λ N * 0 N N * (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ + 2 ∞ λ N * N N * (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ ≤ 2 λ N * 0 √ ρ( 1 γ min + 1 γ * min ) |z| 2 ρ(|z| 2 + ρ 2 ) dρ + 2 ∞ λ N * (M * + 2 √ ρ γ * min ) |z| 2 ρ(|z| 2 + ρ 2 ) dρ ≤ 2( 1 γ min + 1 γ * min ) |z| λ N * /|z| 0 1 √ s(1 + s 2 ) ds + 2M * ∞ λ N * /|z| 1 s(1 + s 2 ) ds + 4 |z| γ * min ∞ λ N * /|z| 1 √ s(1 + s 2 ) ds ≤ 2( 1 γ min + 1 γ * min ) |z| 2 √ λ N * |z| +2M * [ln s √ 1 + s 2 ] ∞ λ N * /|z| + 4 |z| γ * min ∞ 0 1 √ s(1 + s 2 ) ds ≤ 4 λ N * ( 1 γ min + 1 γ * min ) + M * ln(1 + |z| 2 (λ N * ) 2 ) + c u |z| γ * min . Then changing z -iλ N * into z, ln |F N * (z)| ≤ 4 λ N * ( 1 γ min + 1 γ * min ) + M * ln(1 + 2 |z| 2 + λ 2 N * (λ N * ) 2 ) + c u |z| + √ λ N * γ * min ,
which gives (3. 14) with the B m and q m given in (3. 17) and (3. 18).

• Under (2. 1)-(2. 2) and when m > N * , applying the same method, we derive from (3. 8) that

2 ∞ 0 N m (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ ≤ 4( 1 γ min + 2 γ * min ) λ m - 4 γ min λ m -λ N * + M * ln(1 + |z| 2 λ 2 m ) + c u |z| γ * min .
Then changing z -iλ m into z, we obtain (3. 14) with the related B m and q m given in (3. 17) and (3. 18).

• Under (2. 1)-(2. 2) and when m < N * , applying the same method, we derive from (3. 9) that

2 ∞ 0 N m (ρ) ρ |z| 2 |z| 2 + ρ 2 dρ ≤ 8 γ min ( λ m + λ N * -λ m ) + M * ln(1 + |z| 2 max{λ m , λ N * -λ m } 2 ) + c u |z| γ * min .
Then changing z -iλ m into z, we obtain (3. 14) with the related B m and q m given in (3. 17) and (3. 18).

A suitable mollifier.

Motivated by [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], we made in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] the following construction: consider

T > 0, N ≥ 1, a k := C N ,T k 2 with C N ,T := T 2 ∞ k=N 1 k 2 , in order that ∞ k=N a k = T 2 ,
and finally

(3. 19) P N ,T (z) := e iz T 2 ∞ k=N cos(a k z).
Then we have the following

     P N ,T (0) = 1, ∀z ∈ C such that z ≥ 0, |P N ,T (z)| ≤ 1, ∀z ∈ C, |e -iz T 2 P N ,T (z)| ≤ e |z| T 2 .
(2) The behaviour of P N ,T over R: there exist θ 0 > 0, θ 1 > 0, both independent of N and T such that P N ,T satisfies

(3. 21)      C N ,T |x| θ0 1/2 + 1 ≥ N =⇒ ln |P N ,T (x)| ≤ -θ1 2 3 C N ,T |x| θ0 1/2 , C N ,T |x| θ0 1/2 + 1 ≤ N =⇒ ln |P N ,T (x)| ≤ -θ1 (N ) 3 C N ,T |x| θ0 2 .
(3) The behaviour of P N ,T over iR + : there is some constant θ 2 > 0, independent of N and T , such that P N ,T satisfies

(3. 22) ∀x ∈ R + , P N ,T (ix) ≥ e -θ2 √ C N ,T x .
The Proof of Lemma 3.4 follows by elementary analysis techniques. In the following we are going to use the mollifier P N ,T to construct the biorthogonal family. We will make the following choices:

• for T :

(3. 24) T := min{T, 1 (γ * min ) 2 }; • for N : we choose it such that (3. 25) N ≥ 2 + θ 3 (γ * min ) 2 T with a suitable θ 3 (independent of T > 0 and of m ≥ 0, and given in (3. 29)). Then we will prove the following Lemma 3.5. When T and N satisfy (3. 24) and (3. 25), the functions f m,N ,T are entire and satisfy the following properties:

• for all m, n ≥ 1, we have

(3. 26) f m,N ,T (-iλ n ) = δ mn ;
• for all m ≥ 1, for all ε > 0, there exists C m,N ,T ,ε > 0 such that

(3. 27) ∀z ∈ C, |f m,N ,T (-z)e -iz T 2 | ≤ C m,N ,T ,ε e ( T 2 +ε)|z| ; • for all m ≥ 1, f m,N ,T ∈ L 2 (R).
Then we will be in position to apply the Paley-Wiener theorem and to construct the desired biorthogonal sequence. Proof of Lemma 3.5. First, the function f m,N ,T is well-defined since P N ,T > 0 on iR + , and is entire since F m and P N ,T are entire. Next, using (3. 12), we have (3. 26). Next, concerning the exponential type: using (3. but for all ε > 0 we have

C u |z| γ * min = C u ε|z| γ * min √ ε ≤ C 2 u 2(γ * min ) 2 ε + ε 2
|f m,N ,T (x)| ≤ 1 P N ,T (iλ m ) B m q m (|x|)e Cu γ * min √ |x| e - θ 1 8 C N ,T |x| θ 0 1/2 , hence f m,N ,T ∈ L 2 (R) if C u γ * min - θ 1 8 C N ,T θ 0 1/2 < 0,
which is true choosing T and N satisfying (3. 24) and (3. 25): indeed,

C N ,T = T 2 ∞ k=N 1 k 2 , and 1 N = ∞ N 1 y 2 dy ≤ ∞ k=N 1 k 2 ≤ ∞ N -1 1 y 2 dy = 1 N -1 , hence (3. 28) (N -1)T 2 ≤ C N ,T ≤ N T 2 .
Hence, if

(3. 29) (N -1)T > θ 3 (γ * min ) 2 with θ 3 := 2 7 θ 0 C 2 u θ 2 1 ,
we obtain that f m,N ,T ∈ L 2 (R). And one easily verifies that T , N satisfying (3. 24) and (3. 25) satisfy also (3. 29). This completes the proof of Lemma 3.5.

The resulting biorthogonal sequence.

With our choices, the function x → f m,N ,T (-x)e -ixT /2 is in L 2 (R), and we can consider its Fourier transform φ m,N ,T :

φ m,N ,T (ξ) := 1 2π R f m,N ,T (-x)e -ix T 2 e -iξx dx.
It is well-defined since f m,N ,T ∈ L 2 (R), and the Paley-Wiener theorem ([39] p. 100) shows that φ m,N ,T is compactly supported in [-T 2 -ε, T 2 + ε] (thanks to (3. 27)). Since this is true for all ε > 0, φ m,N ,T is compactly supported in [-T 2 , T 2 ]. To obtain good results, we will choose N satisfying the stronger property:

(3. 30) 2 + θ 3 (γ * min ) 2 T ≤ N ≤ 4 + θ 3 (γ * min ) 2 T .
Then we have the following Moreover, it satisfies: there is some universal constant C u independent of T , γ min , γ * min , N * and m such that, for all m ≥ 1, we have This gives (3. 32). Concerning (3. 33), we note that the Parseval equality gives

(3. 33) σ + m,N ,T 2
(3. 34) R |f m,N ,T (x)| 2 dx = R |f m,N ,T (-x)e -ix T 2 | 2 dx = 2π R |φ m,N ,T (ξ)| 2 dξ = 2π T /2 -T /2 |φ m,N ,T (ξ)| 2 dξ. Hence σ + m,N ,T 2
L 2 (0,T ) = e -2λmT T /2 -T /2 |φ m,N ,T (ξ)| 2 dξ = 1 2π e -2λmT R |f m,N ,T (x)| 2 dx.
We need to estimate precisely the last integral. Denote

X N ,T := θ 0 (N -1) 2 C N ,T .
Using (3. 13), (3. 21) and (3. 22), we have

R |f m,N ,T (x)| 2 dx = |x|≤X N ,T |f m,N ,T (x)| 2 dx+ |x|≥X N ,T |f m,N ,T (x)| 2 dx ≤ 2e 2θ2 √ C N ,T λm B 2 m X N ,T 0 q m (x) 2 e 2Cu γ * min √ x e - 2θ 1
(N ) 3 ( C N ,T x θ 0 ) 2 dx + ∞ X N ,T q m (x) 2 e 2Cu γ * min √ x e -2θ 1 2 3 ( C N ,T x θ 0 ) 1/2 dx =: I (<) m + I (>) m .
First we estimate I (<) m ; we denote θ i various constants independent of all the other parameters; we have

X N ,T 0 q m (x) 2 e 2Cu γ * min √ x e - 2θ 1 (N ) 3 ( C N ,T x θ 0 ) 2 dx ≤ q m (X N ,T ) 2 e 2Cu γ * min √ X N ,T ∞ 0 e - 2θ 1
(N ) 3 ( C N ,T x θ 0 ) 2 dx ≤ C q m (X N ,T ) 2 e 2Cu γ * min √ X N ,T (N ) 3/2 C N ,T ≤ C q m (X N ,T ) 2 e 2Cu γ * min √ X N ,T 1 T + 1 (T ) 3/2 γ * min .
Using (3. 24), (3. 28) and (3. 30), we have

X N ,T ≤ θ 4 ( 1 T + 1 (γ * min ) 2 (T ) 2 ), X N ,T γ * min ≤ θ 5 (1 + 1 (γ * min ) 2 T
), and

C N ,T ≤ θ 5 γ * min ; hence I (<) m ≤ c u e θ6 √ λm γ * min B 2 m q m (θ 4 ( 1 T + 1 (γ * min ) 2 (T ) 2 )) 2 e θ 5 (γ * min ) 2 T 1 T + 1 (T ) 3/2 γ * min .
To conclude, we will use the following basic remark: 2 , we obtain that:

y ∈ [0, 1] =⇒ (1 + y) n ≤ 2 n , and y ≥ 1 =⇒ (1 + y) n = y n (1 + 1 y ) n ≤ 2 n y n , hence y ≥ 0 =⇒ (1 + y) n ≤ 2 n (1 + y n ), and a, b ≥ 0 =⇒ (a + b) n ≤ 2 n (a n + b n ). Since (from (3. 24)) 1 T ≤ 1 (γ * min ) 2 (T )
q m (θ 4 ( 1 T + 1 (γ * min ) 2 (T ) 2 )) 2 ≤ 3 + 2 λ 2 m ( 2θ 4 (γ * min ) 2 (T ) 2 ) 2 2M * ≤ 2 2M * 3 2M * + ( 2 λ 2 m ( 2θ 4 (γ * min ) 2 (T ) 2 ) 2 ) 2M * ≤ C 2M * u 1 + ( 1 λ m (γ * min ) 2 (T ) 2 ) 4M * , then ∀m ≥ N * , I (<) m ≤ c u (T ) 3/2 γ * min e θ7 √ λm γ * min e θ 5 (γ * min ) 2 T e θ7 λ N * γ min √ λm C 2M * u 1 + ( 1 λ m (γ * min ) 2 (T ) 2 ) 4M * . Next we estimate I (>) m . Denote L := 2θ 1 2 3 ( C N ,T θ 0 ) 1/2 - 2C u γ * min .
One can easily check that

1 L ≤ C u T γ * min . Then I (>) m = 2e 2θ2 √ C N ,T λm B 2 m ∞ X N ,T q m (x) 2 e 2Cu γ * min √ x e -2θ 1 2 3 ( C N ,T x θ 0 ) 1/2 dx = 2e 2θ2 √ C N ,T λm B 2 m ∞ X N ,T q m (x) 2 e -L √ x dx ≤ 2e θ6 √ λm γ * min e θ7 λ N * γ min √ λm ∞ 0 q m (x) 2 e -L √ x dx = 2e θ6 √ λm γ * min e θ7 λ N * γ min √ λm 2 L 2 ∞ 0 q m ( t 2 L 2 ) 2 e -t dt.
Recalling that

∞ 0 t k e -t dt = k!,
we obtain

I (>) m L 2 4 e -θ6 √ λm γ * min e -θ7 λ N * γ min √ λm ≤ 2 2M * ∞ 0 (3 2M * + 2 (λ m ) 2 L 4 2M * t 8M * )e -t dt = 2 2M * 3 2M * + 2 2M * (8M * )! L 8M * (λ m ) 4M * ≤ C M * u 1 + ( 1 λ m (γ * min ) 2 (T ) 2 ) 4M * (8M * )! .
Finally, we see that there exists some C u independent of m, γ min , γ * min , N * and T such that

σ + m,N ,T 2 L 2 (0,T ) ≤ C u e -2λmT e Cu √ λm γ * min e Cu λ N * γ min √ λm e Cu (γ * min ) 2 T ( 1 (T ) 3/2 + 1 (γ * min ) 2 (T ) 2 ) e CuM * 1 + ( 1 λ m (γ * min ) 2 (T ) 2 ) 4M * (8M * )! ,
ε λn : s ∈ (0, T ) → e -λns , n ≥ 1. It follows from (2. 7) that ∞ n=1 1 λ n < ∞,
and then it is well-known ( [START_REF] Schwartz | Étude des sommes d'exponentielles[END_REF][START_REF] Redheffer | Elementary remarks on completeness[END_REF]) that E(Λ, T ) is a proper subspace of L 2 (0, T ). Moreover, given m ≥ 1, denote Λ m := (λ k ) k =m , and E(Λ m , T ) the smallest closed subspace of L 2 (0, T ) containing the functions ε λ k , with k ≥ 1 and k = m (it does not include ε λm ). Then consider p m the orthogonal projection of ε λm on E(Λ m , T ), and d T,m the distance between ε λm and E(Λ m , T ): we have

(4. 1) d 2 T,m = inf p∈E(Λm,T ) ε λm -p 2 L 2 (0,T ) = T 0 (e -λms -p m (s)) 2 ds.
Then ε λm -p m is orthogonal to E(Λ m , T ) , which implies that ∀n = m, Hence consider

(4. 2) σ - m (s) := e -λms -p m (s) d 2 T,m : the sequence of functions (σ - m ) m≥1 is a biorthogonal family for the set (ε λn ) n≥1 = (e -λnt ) n≥1 in L 2 (0, T ).
Moreover it is optimal in the following sense: if (σ - m ) m≥1 is another biorthogonal family for the set (ε λn

) n≥1 in L 2 (0, T ), then for all m ≥ 1, σ- m -σ - m is orthogonal to all ε λn , hence to E(Λ, T ), hence to σ - m since σ - m ∈ E(Λ, T ). Hence σ- m 2 L 2 (0,T ) = σ - m 2 L 2 (0,T ) + σ- m -σ - m 2 L 2 (0,T ) ≥ σ - m 2 L 2 (0,T ) . Therefore (4. 3) σ- m L 2 (0,T ) ≥ σ - m L 2 (0,T ) = 1 d T,m
.

Hence 1 d T ,m is a lower bound of every biorthogonal sequence (σ - m ) m≥1 ; and a bound from above for d T,m gives a bound from below for every biorthogonal sequence.

At last, we note that if the sequence of functions (σ + m ) m≥1 is a biorthogonal family for the set (e λnt ) n≥1 in L 2 (0, T ), then Hence e -λmT d T ,m is a lower bound of every biorthogonal sequence (σ + m ) m≥1 . In the following (Lemma 4.4), we provide a bound from above for d T,m , that will give a bound from below for every biorthogonal sequence (σ + m ) m≥1 .

4.2.

A general result for sums of exponentials.

Clearly,

d T,m ≤ e -λms -p(s) L 2 (0,T )
for all p ∈ E(Λ m , T ). The idea used in Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] is to chose a particular element p ∈ E(Λ m , T ) in order to provide an upper bound of d T,m . The first thing to note is the following: consider M ≥ m and

q(s) := M +1 i=1 A i e -λis with coefficients A 1 , • • • , A M +1 . Then q ∈ E(Λ m , T
) if and only if A m = 0, and when A m = 0, then

1 A m q(s) = e -λms + m-1 i=1 A i A m e -λis + M +1 i=m+1 A i A m e -λis , hence (4. 5) 1 A m q(s) L 2 (0,T ) ≥ d T,m .
We will choose the coefficients

A 1 , • • • , A M +1 so that q(0) = q (0) = q (0) = • • • = q (M -1) (0) = 0, q (M ) (0) = 1.
The following lemma is essentially extracted from Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF]:

Lemma 4.1. Consider M ≥ 0, and 0 < λ 1 < • • • < λ M +1 . a) There exist coefficients A 1 , • • • , A M +1 so that the function q defined by q(s) := M +1 i=1 A i e -λis satisfies                      q(0) = 0 q (0) = 0 q (0) = 0 . . . q (M -1) (0) = 0 q (M ) (0) = 1.
The coefficients are given by the following formulas:

(4. 6) ∀k ∈ {1, • • • , M + 1}, A k = 1 M +1 i=1,i =k (λ i -λ k ) .
b) With this choice of coefficients, we have

(4. 7) ∀s > 0, 0 < q(s) ≤ s M M ! e -λ1s .
The only difference with Güichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] is the estimate (4. 7) which is more precise than the one obtained in [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF], Lemma 4:

∀s > 0, 0 < q(s) < s M M ! .
In the following, we prove (4. 7), and in a sake of completeness, we give the main arguments for part a) of Lemma 4.1.

Proof of Lemma 4.1. a) We write the linear system

                   0 = q(0) = M +1 i=1 A i 0 = q (0) = M +1 i=1 -λ i A i 0 = q (0) = M +1 i=1 (-λ i ) 2 A i • • • 0 = q (M -1) (0) = M +1 i=1 (-λ i ) (M -1) A i 1 = q (M ) (0) = M +1 i=1 (-λ i ) M A i .
This can be written

(4. 8)        1 1 • • • • • • 1 -λ 1 -λ 2 • • • • • • -λ M +1 (-λ 1 ) 2 (-λ 2 ) 2 • • • • • • (-λ M +1 ) 2 . . . . . . . . . . . . . . . (-λ 1 ) M (-λ 2 ) M • • • • • • (-λ M +1 ) M               A 1 A 2 A 3 . . . A M +1        =        0 0 0 . . . 1       
.

The (M + 1) × (M + 1) matrix A that appears in the left hand side of (4. 8) is invertible: indeed, its determinant is of Vandermonde type, and

det A = k<l (-λ l ) -(-λ k ) = k<l λ k -λ l = 0.
Hence the system (4. 8) is invertible, and the Cramer's formula gives

A k = det B det A ,
where B is the (M + 1) × (M + 1) matrix obtained from A putting the right-hand side member of (4. 8) at the place of the k th -column of A. But then, we can develop det B with respect to the k th -column and we find again a Vandermonde determinant. Then using the formula of Vandermonde determinant, one gets (4. 6). b) We prove (4. 7) by induction. When M = 0, (4. 7) is true. Assume that it is true for some M , and let us prove that it is true for M + 1: take

q(s) := M +2 i=1 A i e -λis ,
where the coefficients A 1 , • • • , A M +2 , are chosen so that

q(0) = q (0) = q (0) = • • • = q (M ) (0) = 0, q (M +1) (0) = 1.
Then the Taylor developments of q and q say that q(s) = s M +1 (M + 1)! + O(s M +2 ) and q (s) = s M M ! + O(s M +1 ) as s → 0. Consider q(s) = e -2λ M +2 s d ds (q(s)e λ M +2 s ). Then q(s) = e -2λ M +2 s d ds

M +2 i=1 A i e (λ M +2 -λi)s = e -2λ M +2 s M +2 i=1 A i (λ M +2 -λ i )e (λ M +2 -λi)s = M +2 i=1 A i (λ M +2 -λ i )e -(λ M +2 +λi)s .
But the last term in the series is clearly equal to 0, hence q is a sum of M + 1 exponentials. Moreover,

q(s) = q (s)e -λ M +2 s + λ M +2 q(s)e -λ M +2 s = ( s M M ! + O(s M +1 ))e -λ M +2 s + λ M +2 ( s M +1 (M + 1)! + O(s M +2 ))e -λ M +2 s = s M M ! + O(s M +1 ) as s → 0.
Hence q(0) = q (0) = q (0) = • • • = q(M-1) (0) = 0, q(M) (0) = 1, and we can apply the induction assumption to q: then 0 < q(s) < s M M ! e -(λ M +2 +λ1)s .

We deduce first that s → q(s)e λ M +2 s is increasing. Since its value in 0 is 0, then q is positive on (0, +∞). Next, we obtain that

d ds (q(s)e λ M +2 s ) ≤ s M M ! e (2λ M +2 -λ M +2 -λ1)s = s M M ! e (λ M +2 -λ1)s ≤ d ds s M +1 (M + 1)! e (λ M +2 -λ1)s ,
hence by integration,

q(s)e λ M +2 s ≤ s M +1 (M + 1)! e (λ M +2 -λ1)s , hence q(s) ≤ s M +1 (M + 1)! e -λ1s ,
which completes the induction argument and the proof of Lemma 4.1.

4.3.

A precise estimate of the remaining part of the exponential function.

It turns out that we will need an estimate for the remaining part of the exponential function

∞ n=N x n n!
in function of x and N . We prove the following general and precise result: Lemma 4.2. We have the following estimates:

(4. 9) ∀N ≥ 1, ∀x ≥ 0, 1 N ! x 1 + x N e x ≤ ∞ n=N x n n! ≤ C 1 N x 1 + x N e x ,
where

C 1 = max x∈R+ (1 -e -x )(1 + x)
x .

Proof of Lemma 4.2. Denote

f N (x) := ∞ n=N x n n! .
Let us prove by induction that

∀N ≥ 0, ∀x ≥ 0, f N (x) ≥ 1 N ! x 1 + x N e x .
First, of course f 0 (x) = e x , and then

f 0 (x) ≥ 1 0! x 1 + x 0 e x .
Next, assume that

∀x ≥ 0, f N (x) ≥ 1 N ! x 1 + x N e x .
We note that f N +1 (x) = f N (x), and d dx

1 (N + 1)! ( x 1 + x ) N +1 e x = 1 (N + 1)! ( x 1 + x ) N e x N + 1 (1 + x) 2 + x 1 + x .
The study of the variations of the function x → N +1 (1+x

) 2 + x 1+x gives ∀x ≥ 0, 1 - 1 4(N + 1) ≤ N + 1 (1 + x) 2 + x 1 + x ≤ N + 1, hence d dx 1 (N + 1)! ( x 1 + x ) N +1 e x ≤ N + 1 (N + 1)! ( x 1 + x ) N e x = 1 N ! ( x 1 + x ) N e x . Then f N +1 (x) = f N (x) ≥ d dx 1 (N + 1)! ( x 1 + x ) N +1 e x ,
and since the values at 0 are 0, we obtain that

∀x ≥ 0, f N +1 (x) ≥ 1 (N + 1)! ( x 1 + x ) N +1 e x .
This proves the first part of (4. 9). For the second part (which is not necessary for us here), we note that

∀x ≥ 0, f 1 (x) ≤ C 1 x 1 + x 1 e x . Assume that ∀x ≥ 0, f N (x) ≤ C 1 N x 1 + x N e x . Then d dx ( x 1 + x ) N +1 e x = ( x 1 + x ) N e x N + 1 (1 + x) 2 + x 1 + x ≥ (1- 1 4(N + 1) )( x 1 + x ) N e x .
Hence

f N +1 (x) = f N (x) ≤ C 1 N x 1 + x N e x ≤ C 1 N 1 - 1 4(N +1) d dx ( x 1 + x ) N +1 e x .
To conclude, note that

∀N ≥ 1, N 1 - 1 4(N +1) ≤ N + 1 : indeed, (N + 1)(1 - 1 4(N + 1) ) = N + 1 - 1 4 = N + 3 4 ≥ N.
Hence, we obtain that

∀x ≥ 0, f N +1 (x) ≤ C 1 (N + 1) x 1 + x N +1 e x ,
which concludes the induction, and the proof of (4. 9). 4.4. Consequence: a bound from above for the distance d T,m .

As a consequence of the upper estimate (4. 5) for the distance and of Lemma 4.1, we obtain the following inequality: for all m ≥ 1, for all M ≥ m, we have

(4. 10) d T,m ≤ M +1 i=1,i =m |λ i -λ m | T 0 s 2M M ! 2 e -
* := [ 2 √ λ 1 γ max ] + m + 2 and C(T, γ max , λ 1 , m) = 6 √ 1 + 2T λ 1 π 2 √ 2T (k * -1)! (m + k * + 3)! (m -1)! (T γ 2 max ) k * +2 (1 + T γ 2 max ) m+k * +3 . Then (4. 11) ∀m ≥ 1, 1 d T,m ≥ C(T, γ max , λ 1 , m) e 1 T γ 2 max . Proof of Lemma 4.3. Of course T 0 s 2M M ! 2 e -2λ1s ds ≤ T 2M +1 M ! 2 (2M + 1)
, an, on the other hand,

T 0 s 2M M ! 2 e -2λ1s ds ≤ T 2M M ! 2 T 0 e -2λ1s ds ≤ T 2M M ! 2 1 -e -2λ1T 2λ 1 . Hence T 0 s 2M M ! 2 e -2λ1s ds ≤ T 2M M ! 2 inf{ T 2M + 1 , 1 -e -2λ1T 2λ 1 }. But it is easy to check that ∀a, b > 0, inf{a, 1 b } ≤ 2a 1 + ab .
Indeed, inf{a, 1 b } = a if ab ≤ 1, and in this case 1 + ab ≤ 2, hence a(1 + ab) ≤ 2a. On the other hand, when ab ≥ 1, inf{a, 1 b } = 1 b , and 1 + ab ≤ 2ab. We deduce that (4. 12)

T 0 s 2M M ! 2 e -2λ1s ds 1/2 ≤ T M M ! √ 2T √ 2M + 1 + 2T λ 1 .
Now it remains to estimate the product

M +1 i=1,i =m |λ i -λ m | = M +1 i=1,i =m | λ i -λ m | M +1 i=1,i =m ( λ i + λ m ) .
We derive from (2. 7) first that

| λ i -λ m | ≤ γ max |i -m|,
and next that

λ i + λ m ≤ 2 λ 1 + γ max (i + m). Hence • first M +1 i=1,i =m ( λ i + λ m ) ≤ M +1 i=1,i =m (2 λ 1 + γ max (i + m)) ≤ γ M max (M + 1 + [ 2 √ λ1 γmax ] + m + 1)! ([ 2 √ λ1 γmax ] + m + 1)! = c (+) γ M max (M + k * )! with c (+) = 1 ([ 2 √ λ1 γmax ] + m + 1)! and k * := [ 2 √ λ 1 γ max ] + m + 2; • next M +1 i=1,i =m | λ i -λ m | ≤ M +1 i=1,i =m γ max |i -m| = γ M max (m -1)!(M -(m -1))! = c (-) γ M max (M -(m -1))! with c (-) = (m -1)!.
Combining this with (4. 12), we derive from (4. 10)

d T,m ≤ c (+) c (-) √ 2T √ 2M + 1 + 2T λ 1 (T γ 2 max ) M (M + k * )! (M -m + 1)! M ! . Denote c * := c (+) c (-) √ 2T √ 1 + 2T λ 1 .
Then, to conclude, we note that

1 d T,m = 6 π 2 ∞ M =m+1 1 (M -m) 2 1 d T,m ≥ 6 π 2 c * ∞ M =m+1 1 (M -m) 2 M ! (M + k * )! (M -m + 1)! ( 1 T γ 2 max ) M ≥ 6 π 2 c * ∞ M =m+1 1 (M + k * + 2)! ( 1 T γ 2 max ) M = 6 π 2 c * (T γ 2 max ) k * +2 ∞ n=m+k * +3 1 n! ( 1 T γ 2 max ) n .
And using Lemma 4.2, we obtain that (4. 11). This gives the expected exponential behaviour in 1/(T γ 2 max ). In the following we take care of the asymptotic gap γ * max .

4. 13) 1 d T,m ≥ b * (T, γ max , γ * max , N * , λ 1 , m) e 1 T (γ * max ) 2
where b * is given by

• when m ≤ N * , we have (4. 14) b * (T, γ max , γ * max , N * , λ 1 , m) = C * √ 1 + T λ 1 √ T (T (γ * max ) 2 ) K * +K * +2 (1 + (T (γ * max ) 2 )) N * +K * +K * +3
, where

C * = c u (γ * max ) 2(N * -1) C (+) C (-) 1 (N * + K * + K * + 3)! ,
and C (+) , C (-) , K * and K * are given respectively in (4. 17), (4. 20), (4. 18) and (4. 21); • when m > N * , we have This concludes the proof of Lemma 4.4 when m > N * .

(4. 15) b * (T, γ max , γ * max , N * , λ 1 , m) = C * √ 1 + T λ 1 √ T (T (γ * max ) 2 ) K * +2 (1 + T (γ * max ) 2 ) m+K * +3 , where C * = c u C(+) C(-) 1 (m + K * + 3

1 . Introduction 1 . 1 .

 111 Presentation of the subject.

(1. 4) σ m 2 L 2

 22 (0,T ) ≤ B m e -2λmT e Cu T (γ min ) 2 e Cu √ λm γ min , with an explicit value of B m = B m (T, γ min , m) (rational in T ).

(1. 7 ) σ m 2 L 2 2 T

 7222 (0,T ) ≤ B * m e -2λmT e C 0 > 0 is a universal constant and B * m (T, γ min , γ * min , N * , m) is a rational function of T that we determine explicitly. Let us observe that the presence of the exponential factors e

3 . 1 3. 1 .

 311 Proof of Theorem 2.The general strategy.

  The main point in (3. 3)-(3. 6) is to observe that N n behaves as √ ρ γ * min as ρ → +∞, and precising the needed additionnal constants. Proof of Lemma 3.1. Take k > n. Then

Lemma 3 . 3 .,Remark 3 . 3 .

 3333 a) Assume that the gap assumption (2. 1) is satisfied. Then the function F m satisfies the following growth estimate: there is some uniform constant C u (independent of m, γ min , and z) such that (3. 13) ∀z ∈ C, |F m (z)| ≤ e that the gap assumptions (2. 1)-(2. 2) are satisfied. Then the function F m satisfies the following growth estimate: there is some uniform constant C u (independent of m, γ min , γ * min and z), such that (3. 14) ∀m ≥ 1, ∀z ∈ C, |F m (z)| ≤ B m q m (|z|) e where M * has been defined in (2. 3). The main point in Lemma 3.3 is to obtain estimates of the growth of F m in e Cu √ |z| γ * min under (2. 1)-(2. 2), with explicit constants (given in (3. 15) and (3. 16)), that will help us in the following. Comparing with (3. 13), this gives a better idea of the improvement brought by 'large' gap γ * min and of the price to pay due to the 'small' gap γ min for the N * first eigenvalues. In fact we will first prove the following better estimates: (3. 14) holds true with(3. 17)

Lemma 3 . 4 . 1 )

 341 ([6]) (The regularity and the growth of P N ,T over C: The function P N ,T is entire over C and satisfies(3. 20)

3. 5 .

 5 A sequence of holomorphic functions satisfying (3. 1). Consider (3. 23) ∀m ≥ 0, ∀z ∈ C, f m,N ,T (z) := F m (z) P N ,T (-z) P N ,T (iλ m ) .

2 | 1 P 1 P

 211 14) and (3. 20), we have |f m,N ,T (-z)e -iz T 2 | = |F m (-z)| |P N ,T (z)e -iz T 2 | |e -iz T -T N ,T (iλ m ) , ≤ N ,T (iλ m ) B m q m (|z|)e Cu √ |z| γ * min e |z| T 2 e |z| T -T 2 ;

Lemma 3 . 6 . 2 -

 362 Take T and N satisfying (3. 24) and (3. 30), and consider (3. 31) σ + m,N ,T (t) := φ m,N ,T ( T t)e -λmT .Then the family (σ + m,N ,T ) m≥1 is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, ,T (t)e λnt dt = δ mn .

L 2 ( 2 -T / 2 φ 2 -T / 2 φ 2 T / 2 -T / 2 φ

 22222222 0,T ) ≤ C u e -2λmT e Cu √ λm γ * min B(T, γ min , γ * min , N * , m), where B(T, γ min , γ * min , N * , m) is given by (2. 6). Proof of Lemma 3.6. The Fourier inversion theorem gives that f m,N ,T (-x)e -ix T 2 = R φ m,N ,T (ξ)e iξx dξ = T /m,N ,T (ξ)e iξx dξ. e -λmT e λnt dt = e -λmT T /m,N ,T (ξ)e λn( T 2 -ξ) dξ = e -λmT e λn T m,N ,T (ξ)e -λnξ dξ = e -λmT e λn T 2 f m,N ,T (-iλ n )e λn T 2 = f m,N ,T (-iλ n )e (λn-λm)T = δ mn .

which gives ( 3 . 33 ) 4 . 2 4. 1 .

 333421 and completes the proof of Lemma 3.6 and of Theorem 2.1. Proof of Theorem 2.A lower bound for any biorthogonal family. Denote E(Λ, T ) the smallest closed subspace of L 2 (0, T ) containing the functions

T 0 (

 0 e -λms -p m (s))e -λns ds = 0, and T 0 (e -λms -p m (s))e -λms ds = T 0 (e -λms -p m (s))(e -λms -p m (s)) ds = d 2 T,m .

  -s)e λmT e -λns ds = δ mn , hence (σ + m (T -s)e λmT ) m is biorthogonal for the set (e -λnt ) n≥1 in L 2 (0, T ). This implies that (4. 4) σ+ m L 2 (0,T ) ≥ e -λmT d T,m .

(

( 2 λ 1 ( 2 .( 2 λ 1 2 √γ)e 1 /

 2122121 )! where C(+) , C(-) and K * are given respectively in (4. 23), (4. 25) and (4. 18).Proof ofLemma 4.4. The starting point is of course (4. 10) and (4. 12). Concerning the estimate of the product, we proceed in the same way as previously, distinguishing several cases. a) First we investigate what can be said when m ≤ N * < M + 1: in this case,• first we see that∀i ≥ N * + 1, λ i + λ m = λ i -λ N * + λ N * + λ m ≤ γ * max (i -N * ) + 2 λ 1 + (m + N * )γ max ; hence (4. 16) M +1 i=1,i =m ( λ i + λ m ) = N * i=1,i =m ( λ i + λ m ) λ 1 +(m+N * )γ max +γ * max (i-N * )) ≤ C (+) (γ * max ) M (M + 1 -N * + [ 2 √ λ 1 + (N * + m)γ max γ * max ] + 1)! = C (+) (γ * max ) M (M + K * )! with (4. 17) C (+) = ( γ max γ * max ) N * -1 (N * + m + [ (i -N * ) + γ max (N * -m) ≤ C (-) (γ * max ) M (M -N * + 2 + [ γ max γ * max (N * -m)])! = C (-) (γ * max ) M (M + K * )-m)] -N * + 2;We deduce from (4. 10), (4. 12), (4. 16) and (4. 24) thatd T,m ≤ C (+) C (-) M + K * )! (M + K * )! M ! (T (γ * max ) 2 ) M .DenoteC * := C (+) C (-) √ 2T √ 1 + 2T λ 1 . Hence d T,m ≤ C * (M + K * )! (M + K * )! M ! (T (γ * max ) 2 ) M .Then, as we did before,K * )! (M + K * )! = 1 (M -N * ) 2 1 (M + K * )! (M + 1) • • • (M + K * ) ≥ 1 (M + K * + K * + 2)! . K * + K * + 2)! ( 1 T (γ * max ) 2 ) M = 6 π 2 C * (T (γ * max ) 2 ) K * +K * +2 ∞ n=N * +K * +K * +3 ( + K * + K * + 3)! X K * +K * +2 (1 + X) N * +K * +K * +3 e 1/X .with X = T (γ * max ) This concludes the proof of Lemma 4.4 when m ≤ N * . b) In the same way, if m > N * , we have • first M +1 i=1,i =m( λ i + λ m ) = N * i=1 ( λ i + λ m ) M +1 i=N * +1,i =m ( λ i + λ m ) +γ max (i+m)) M +1 i=N * +1,i =m (2 λ 1 +(m+N * )γ max +γ * max (i-N * )) ≤ (γ max ) N * (N * + m + [ λ1+(N * +m)γmax γ * max ] + 1)!(m -N * + [ 2 √ λ1+(N * +m)γmax γ max |i -m| M +1 i=N * +1,i =m γ * max |i -m| = (γ max ) N * (γ * max ) M -N * (m-1)! (M + 1 -m)! = C(-) (γ * max ) M (M + 1 -N * (m -1)!;• then we can conclude:d T,m ≤ C * (M + K * )! (M + 1 -m)! M ! T M (γ * max ) (T (γ * max ) 2 ) .

  2λ1s ds

	1/2

.

It remains to estimate the terms that appear in the right hand side. This is the object of the next sections, and it is based on the gap conditions (2. 7) and (2. 8).

4.4.a. Estimate under the uniform gap condition (2. 7).

We prove the following: Lemma 4.3. Assume that (λ n ) n satisfies (2. 7). Denote k

  4.4.b. Estimate under the uniform gap condition (2. 7) and the asymptotic gap condition (2. 8). Now, taking into account the 'asymptotic gap' given by (2. 8), we will be able to improve the previous estimate, roughly speaking replacing γ 2 max by (γ * max ) 2 in the exponential factor. Lemma 4.4. Assume that (λ n ) n satisfies (2. 7)-(2. 8). Then

Acknowledgements. The authors thank the anonymous referee for all the comments on this paper, and in particular for having attracted their attention on the cases that are not covered by their gap conditions (in particular the example {n 2 , n 2 + √ n}), and the possible generalization (1. 8).

This funds provided by the national group GNAMPA and the GDRE CONEDP.


Genni Fragnelli

Patrick Martinez

Judith Vancostenoble

A new age-dependent population model with diffusion and gestation processes

Keywords: population dynamics, coupled equations, qualitative properties, asymptotic behaviour Mathematics Subject Classification (2010): 35Q92, 35B40, 35B51

.

-Introduction

The goal of this paper is to introduce a new age-structured population model with diffusion and gestation processes and to make a complete study of the qualitative properties of its solutions. The model we suggest here is inspired by a previous one that has been introduced in [START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF] and studied in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF][START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF].

Classically, most of the population models proposed before (see for example [START_REF] Anit ¸a | Analysis and control of age-dependent population dynamics[END_REF][START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF][START_REF] Piazzera | An age-dependent population equation with delayed-birth process[END_REF][START_REF] Pozio | Some conditions for global asymptotic stability of equilibria of integro-differential equations[END_REF][START_REF] Wu | Theory and Applications of Partial Functional Differential Equations[END_REF] and the references therein) consider only a delay due to pregnancy but they do not take into account that a lot of things can happen during the time of gestation (for example the pregnant individuals may move, die or bear before the fixed time). Those "classical" models of population dynamics with diffusion may in general be viewed as delay equations and studied using this theory.

The model introduced in [START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF] is more realistic since it takes into account for example the fact that, in general, pregnant individuals move during the period of gestation and therefore can bear in a place different from that they were fecundated. It is a system of two partial differential equations whose unknowns are the size of the total population and the size of the population of individuals in gestation. Of course these two populations are connected, and the novelty of the present paper will be precisely a new coupling relation between these populations. The model introduced in [START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF] has been studied in several papers, with different assumptions and methods:

• using the theory of delay equations with nonautonomous past, see [START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF] and also [START_REF] Fragnelli | Delay Equations with Nonautonomous Past[END_REF][START_REF] Fragnelli | Partial functional differential equations with nonautonomous past in L p -spaces[END_REF][START_REF] Fragnelli | A spectral mapping theorem for semigroups solving PDEs with nonautonomous past in L p -spaces[END_REF];

• in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF], we completed the results of [START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF], providing a complete analysis of the following questions: existence, positivity, asymptotic behavior of the solutions, under rather natural biological assumptions (and moreover with direct methods: Fourier decomposition, maximum principles).

However, in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] we also pointed out some weakness of the model. Indeed we investigated another natural qualitative property that was not studied before in [START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF][START_REF] Fragnelli | A population equation with diffusion[END_REF] : is the number of individuals in gestation less than the total number of individuals (assuming of course that this property is satisfied at the initial time) ? Surprisingly, the answer was not obvious, the result was even false in general, and the natural and expected property was true when the initial data and parameters satisfy some additional, quite strong, assumption meaning, roughly speaking, that the property is true (and thus the model is valid) if the population of pregnant individuals is "small" with respect to the total population. The goal of the present work is to introduce a new model that improves the previous one in the sense that the property is true without extra assumption.

In section 2 we introduce the new model we are interested in. In section 3, we give our main results on it. Section 4 contains some comments and comparison with other connected models. Section 5 contains the proofs of our main results. Section 6 is devoted to some remarks and perspectives.

-The new population model

We consider a spatially distributed population where individuals are characterized by their position. (No sex or age differences are allowed). A special attention is paid to the mechanism of pregnancy, that leads to a delay in the replacement of the population. The model wants to take into account the events that may happen during the gestation : the pregnant individuals may move, die or bear before the term. For this reason, within the total population, we distinguish pregnant individuals and we refer to them by considering their 'age of gestation' a, ranging in [0, r] where r > 0 is fixed.

In general, individuals are supposed to die at a given death rate d whereas the pregnant individuals are supposed to die at a death rate d , with d ≥ d. Pregnant individuals are also supposed to bear according to a rate b = b(a). Moreover, we assume that the dispersal of the population through the environment is realized by the Laplace operator.

Summing up, let Ω ⊂ R n be open, connected and bounded with smooth boundary, and denote, respectively, by u(t, x) and v(t, a, x) the total population at time t and position x and the subpopulation of pregnant individuals at time t and position x with a time of gestation a; then the dynamics of the two populations is governed by the following equations :

(1)

u t (t, x) -∆u(t, x) + du(t, x) = r 0 b(a)v(t, a, x) da, v t (t, a, x) + v a (t, a, x) -∆v(t, a, x) + d v(t, a, x) = -b(a)v(t, a, x),
where (t, a, x) ∈ R * + × (0, r) × Ω. In the second equation, the term b(a)v(t, a, x) represents the density of pregnant individuals that bear at time t, at place x and after a time of gestation denoted by a. The minus sign comes from the fact that the pregnant individual that bears is no more pregnant, and hence this term behaves like a mortality one. In the first equation, the coupling term r 0 b(a)v(t, a, x) da represents the related increase of the total population (i.e. the newborn individuals) at time t and place x.

We consider here the following homogeneous Dirichlet boundary conditions

(2) u(t, x) |∂Ω = 0, t > 0, x ∈ ∂Ω, v(t, a, x) |∂Ω = 0, t > 0, (a, x) ∈ (0, r) × ∂Ω,
which means that no individual reaches the borderline. (Notice that we also could consider homogeneous Neumann boundary conditions which means that no individual crosses the borderline). Moreover the initial population at time t = 0 is given by

(3) u(0, x) = u 0 (x) ≥ 0, x ∈ Ω, v(0, a, x) = v 0 (a, x) ≥ 0, (a, x) ∈ (0, r) × Ω.
Finally, we need to describe how the number of fecundated individuals depends on the total population, which gives another coupling condition. In [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF][START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF], the assumption is that the number of fecundated individuals at time t and place x is a fixed proportion of the whole population present at that place and time :

(4) v(t, 0, x) = f 0 u(t, x), t > 0, x ∈ Ω.
In the present paper, we are going to assume that the population v(t, 0, x) that is fecundated at time t and place x is a fraction of the total population that has not yet been fecuntaded at time t and place x. To that purpose, we consider (5) V (t, x) := r 0 v(t, a, x) da, which represents the total population of pregnant individuals at time t and place x, and we replace (4) by ( 6) v(t, 0, x) = f 0 (u(t, x) -V (t, x)), t > 0, x ∈ Ω, which means that the population that becomes pregnant at time t and place x, which is v(t, 0, x), is a fraction of the total population that has not yet been fecuntaded at time t and place x, which is u(t, x) -V (t, x). This lead us to consider the following new model :

(7)              u t (t, x) -∆u(t, x) + du(t, x) = r 0 b(a)v(t, a, x) da, v t (t, a, x) + v a (t, a, x) -∆v(t, a, x) + d v(t, a, x) = -b(a)v(t, a, x), v(t, 0, x) = f 0 u(t, x) - r 0 v(t, a, x) da , u(t, x) |∂Ω = 0, v(t, a, x) |∂Ω = 0, u(0, x) = u 0 (x), v(0, a, x) = v 0 (a, x),
where (t, a, x) ∈ R * + × (0, r) × Ω. Note that there is a nonlocal local term in the first equation and now also in the (coupling) boundary condition.

-Main results

-Assumptions and well-posedness

Let us make the following assumptions:

(8) r > 0, d ≥ d > 0 and f 0 > 0, (9) b ∈ L ∞ loc ([0, r)) such that b ≥ 0, b nondecreasing, r 0 b(a) da = +∞, (10)
u 0 ∈ L 2 (Ω), u 0 ≥ 0, v 0 ∈ L 2 ((0, r) × Ω), v 0 ≥ 0 such that √ bv 0 ∈ L 2 ((0, r) × Ω).
In the following, we set

∀s ∈ (0, r), b(s) := s 0 b(σ) dσ.
For any characteristic line

S := {(t, a) ∈ (0, T )×(0, r) | a-t = a 0 -t 0 } = {(t 0 +s, a 0 +s) | s ∈ (0, r-a 0 )}, with (t 0 , a 0 ) ∈ (0, T ) × {0} ∪ {0} × (0, r), we denote by W 1,1 (S; L 2 (Ω)) the space of functions v : S → L 2 (Ω) such that v(t 0 + •, a 0 + •) : (0, r -a 0 ) → L 2 (Ω) is in W 1,1 ((0, r -a 0 ); L 2 (Ω)
). Then we have the following

T h e o r e m 3.1. Well-posedness. Assume (8), ( 9), [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] . Then for any T > 0, problem (7) has a unique solution (u, v) on (0, T ) such that

(11) u ∈ C([0, T ]; L 2 (Ω)) ∩ W 1,1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)) ∩ L 2 loc (0, T ; H 2 (Ω)), ( 12
) v ∈ C( S; L 2 (Ω)) ∩ W 1,1 (S; L 2 (Ω)) ∩ L 2 (S; H 1 0 (Ω)) ∩ L 2 loc (S; H 2 (Ω)),
for almost any characteristic line S of the equation a -t = a 0 -t 0 ,

√ bv ∈ L 2 ((0, T )×(0, r)×Ω) and r 0 b(a)v(•, a, •) da ∈ L 2 ((0, T )×Ω). (13)
Moreover,

(14) ∀ t, v(t -ε, r -ε, •) → 0 in L 2 (Ω) as ε → 0.

-Qualitative properties

T h e o r e m 3.2. Qualitative properties. Assume (8), ( 9), ( 10) are satisfied.

a) Assume also that

(15) V 0 (x) ≤ u 0 (x) in Ω.
Then the solution (u, v) of ( 7) satisfies:

(16) u ≥ 0 in (0, T ) × Ω and v ≥ 0 in (0, T ) × (0, r) × Ω, and (17)
∀ t ≥ 0, V (t, •) ≤ u(t, •) in Ω. b) Assume that there is some θ ∈ [0, 1) such that (18) V 0 (x) ≤ (1 -θ)u 0 (x) in Ω and d -d ≥ f 0 θ 1 -θ .
Then the solution (u, v) of ( 7) satisfies ( 16) and

(19) ∀ t ≥ 0, V (t, •) ≤ (1 -θ)u(t, •) in Ω.

-Asymptotic behaviour

Finally we set

(20) d = d + f 0 , d = 1 2 (d + d ), f0 = 2f 0 , b = b + 1 2 (d -d).
We also denote by λ 0 the smallest eigenvalue of the Laplace operator with Dirichlet boundary conditions and

Then we prove the following

T h e o r e m 3.3. Asymptotic behaviour. Assume (8), ( 9), [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] . Assume that (u 0 , v 0 ) = (0, 0) (otherwise the solution is identically equal to zero). Then the solution of ( 7) satisfies (i) if R0 < d + λ 0 , then u goes to zero exponentially fast in L 2 (Ω) as t → ∞, and v goes to zero exponentially fast in L 2 ((0, r) × Ω).

(ii) if R0 = d + λ 0 , then u converges exponentially fast (in L 2 (Ω)) to some stationary state u * ∈ L 2 (Ω) as t → ∞, and v goes to some stationary state v * ∈ L 2 ((0, r) × Ω).

(iii) if R0 > d + λ 0 , then u goes exponentially fast to infinity in L 2 (Ω)-norm as t → ∞, and v goes exponentially fast to infinity in L 2 ((0, r) × Ω)-norm.

-Comparison with the existing literature

In this section, we compare the new model [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF] to the existing literature. First let us recall the results obtained in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] concerning the model

(22)                u t (t, x) -∆u(t, x) + du(t, x) = r 0 b(a)v(t, a, x) da, v t (t, a, x) + v a (t, a, x) -∆v(t, a, x) + d v(t, a, x) = -b(a)v(t, a, x), v(t, 0, x) = f 0 u(t, x), u(t, x) |∂Ω = 0, v(t, a, x) |∂Ω = 0, u(0, x) = u 0 (x), v(0, a, x) = v 0 (a, x),
where (t, a, x) ∈ R * + ×(0, r)×Ω. This will be useful to understand the differences with [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF], and also to prove Theorems 3.1, 3.2 and 3.3.

-Well-posedness of the model (22)

T h e o r e m 4.1. (Well-posedness, [10, Theorem 2.1, Theorem 2.2]) Assume (8), ( 9) and [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]. For any T > 0, problem (22) has a unique solution (u, v) on (0, T ) such that

(23) u ∈ C([0, T ]; L 2 (Ω)) ∩ W 1,1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)) ∩ L 2 loc (0, T ; H 2 (Ω)), (24) v ∈ C( S; L 2 (Ω)) ∩ W 1,1 (S; L 2 (Ω)) ∩ L 2 (S; H 1 0 (Ω)) ∩ L 2 loc (S; H 2 (Ω)),
for almost any characteristic line S of equation a -t = a 0 -t 0 ,

√ bv ∈ L 2 ((0, T ) × (0, r) × Ω) and r 0 b(a)v(a) da ∈ L 2 ((0, T ) × Ω). (25)
Moreover,

(26) ∀ t, v(t -ε, r -ε, •) = 0 in L 2 (Ω) as ε → 0.
Remark. Note that, in [START_REF] Fragnelli | A population equation with diffusion[END_REF] and [START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF], b is assumed to belong to L ∞ (0, r) or to L 1 (0, r) so that some associated delay operator is well-defined. In [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF], we prefered to assume

(27) r 0 b(a) da = +∞.
Indeed it is a more "natural" assumption that ensures v(•, r, •) = 0, which means that r is the maximal time of gestation. But this induces several technical difficulties, and the well-posedness of ( 22) was proved when the initial conditions belong to some suitable weighted space (see assumption [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]).

-Qualitative properties and weakness of the model (22)

T h e o r e m 4.2. (Qualitative properties, [10, Theorem 2.2]) Assume (8), ( 9) and [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]. Then the solution (u, v) of (22) satisfies:

(28) u ≥ 0 a.e. on (0, T ) × Ω and v ≥ 0 a.e. on (0, T ) × (0, r) × Ω.

Moreover, if (29)
f 0 u 0 (x) r 0 e -b(τ ) e -(d -d)τ dτ ≤ u 0 (x) -V 0 (x)
for almost all x ∈ Ω. Then the solution satisfies

(30) ∀ t ≥ 0, ∀ x ∈ Ω, u(t, x) ≥ V (t, x).
The main novelty in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] was the study of property (30), that means that the total population u(t) is larger than the total population of pregnant individuals V (t). Of course, in order to ensure such a property, it is necessary to assume that it is satisfied at time t = 0. The natural assumption in order to get (30) would simply be the following one: u 0 ≥ V 0 . Observe that our assumption (29) implies u 0 ≥ V 0 but is unfortunately stronger. Note also that, even if the result is very natural, it does not follow easily from the equations and the strong assumption (29) is really needed to prove it. Indeed it is possible to construct a counter-example to this property (see [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]). Moreover, even under condition (29), the proof of (30) was not obvious since we had to compare u with V whereas we did not have a system of equations in (u, V ) but in (u, v).

Hence the previous model ( 22) is realistic only when the pregnant individuals population is "small enough". This comes from the fact that the model (22) assumes that the number of fecundated individuals at time t and place x is a fixed proportion of the whole population present at that place and time (remember ( 4)). This coupling condition is natural when the pregnant individuals population is "small". On the contrary, if the pregnant individuals population is "large", this cannot be a realistic model. This is the motivation for the more realistic condition [START_REF] Anit ¸a | Optimal impulse-control of population dynamics with diffusion[END_REF]. And the gain is that the new model effectively corrects the weakness of the first one since property ( 17) is now valid without restriction on the parameters and initial data.

-Asymptotic behaviour for the model (22)

Finally, in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF], we also completely studied the asymptotic behavior of the solutions of ( 22): denote λ 0 the smallest eigenvalue of the Laplace operator with Dirichlet boundary conditions, ϕ 0 the associated eigenfunction, and

R 0 := f 0 r 0 b(s)e -( b(s)+(d +λ0)s) ds.

Then we proved the following

T h e o r e m 4.3. (Asymptotic behaviour, [10, Theorem 2.3]) Assume (8), ( 9) and [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]. Assume that (u 0 , v 0 ) = (0, 0) (otherwise the solution is identically equal to zero). Then the solution of (22) satisfies (i) if R 0 < d + λ 0 , then u goes to zero exponentially fast as t → ∞.

(ii) if R 0 = d + λ 0 , then u converges exponentially fast to a stationary state.

(iii) if R 0 > d + λ 0 , then u goes exponentially fast to infinity as t → ∞.

Remarks. The stationary state in case (ii) and the rates of convergence in each cases may be explicitly determined by the proof. (We refer to [10, Theorem 2.3] for the detailed statement of Theorem 4.3). Concerning the behavior of v, it is then easy to see that v(t, •, •) goes exponentially fast (in L 2 ((0, r) × Ω)) to zero, to a stationary state or to infinity as t → ∞ respectively if

R 0 < d + λ 0 , R 0 = d + λ 0 or R 0 > d + λ 0 .

-Proofs of Theorems 3.1, 3.2 and 3.3

To prove Theorems 3.1, 3.2 and 3.3, we will make analogies with (22). In particular, there is a change of unknowns that allows us to transform (7) into (22). This is not sufficient to conclude, as we will explain in the following, but it helps a lot.

As in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] we work under the assumption (27) on b. This is natural from a biological point of view, since it implies that the solutions of problem (22) satisfy the property (26), and will imply a similar property for the solutions of (7). Assume (u, v) is a solution of ( 7) such that v(a = r) = 0. Then, it is easy to see that

(31) (û, v) = u -V 2 , v
solves a system having the form of ( 22). More precisely, we have

(32)            ût (t, x) -∆û(t, x) + dû(t, x) = r 0 b(a)v(t, a, x) da, vt (t, a, x) + va (t, a, x) -∆v(t, a, x) + d v(t, a, x) = -b(a)v(t, a, x), v(t, 0, x) = f0 û(t, x), û(t, x) |∂Ω = 0, v(t, a, x) |∂Ω = 0, û(0, x) = û0 (x), v(0, a, x) = v0 (a, x), with (33)
     d = d + f 0 , d = 1 2 (d + d ), f0 = 2f 0 , b = b + 1 2 (d -d), û0 = 1 2 (u 0 -V 0 ), v0 = v 0 . 5.1.2 -From (22) to (7)
Reciprocally, assume (û, v) is a solution of (32) such that v(a = r) = 0. Then, it is easy to see that

(34) (u, v) = 2û + r 0 vda, v solves (7), with (35)
         d = d -1 2 f0 , d = 2 d -d + 1 2 f0 , f 0 = 1 2 f0 , b = b -( d -d + 1 2 f0 ), u 0 = 2û 0 + r 0 v0 da, v 0 = v0 .

-Proof of Theorem 3.1

Assume that the parameters d, d , f 0 , b, u 0 , v 0 satisfy (8), ( 9), ( 10) and [START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF]. Assume that we have a solution (u, v) of ( 7) such that v(a = r) = 0.

Then apply the transform (31). This transforms problem ( 7) into (32) with the parameters (33). The assumptions of Theorem 4.1 are all satisfied, except the hypothesis d ≥ d that is not always satisfied. But this is not essential, since looking to the proof of Theorem 4.1 one can check that the result of Theorem 4.1 remains true without this assumption. Indeed, the existence of the solution of (32) follows from a fixed point argument: consider

L 2
+ ((0, T ) × Ω) := { ĥ ∈ L 2 ((0, T ) × Ω) | ĥ is nonnegative}; let ĥ be given in L 2 + ((0, T ) × Ω) and consider the problem

(36)                    ût (t, x) -∆û(t, x) + dû(t, x) = r 0 b(a)v(t, a, x) da, vt (t, a, x) + va (t, a, x) -∆v(t, a, x) + d v(t, a, x) = -b(a)v(t, a, x), v(t, 0, x) = f0 ĥ(t, x), û(t, x) |∂Ω = 0, v(t, a, x) |∂Ω = 0, û(0, x) = û0 (x), v(0, a, x) = v0 (a, x).
From the classical results concerning the heat equation, there exists a unique solution (û (h) , v(h) ) of (36). Moreover, the map

F : L 2 + ((0, T ) × Ω) → L 2 + ((0, T ) × Ω) ĥ → û(h) ,
where (û (h) , v(h) ) is the solution of (36) associated to ĥ, is a contraction on the space L 2 + ((0, T ) × Ω), when L 2 ((0, T ) × Ω) is endowed with the norm ∀q ∈ L 2 ((0, T ) × Ω), q λ := e -λt/2 q L 2 ((0,T )×Ω) ,

where the constant λ > 0 is large enough, and this does not depend on the sign of dd (we refer to [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF] for the detailed proof). Then the proof of Theorem 4.1 holds true without any assumption on the sign of dd, and (32) has a unique solution (û, v) on (0, T ) such that

û ∈ C([0, T ]; L 2 (Ω)) ∩ W 1,1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)) ∩ L 2 loc (0, T ; H 2 (Ω)), v ∈ C( S; L 2 (Ω)) ∩ W 1,1 (S; L 2 (Ω)) ∩ L 2 (S; H 1 0 (Ω)) ∩ L 2 loc (S; H 2 (Ω)), for almost any characteristic line S of equation a -t = a 0 -t 0 , bv ∈ L 2 ((0, T ) × (0, r) × Ω) and r 0 b(a)v(a) da ∈ L 2 ((0, T ) × Ω).
And (26) remains also true: thanks to (8)-( 9), we have that b

∈ L ∞ loc ([0, r)), b ≥ 0, b is nondecreasing, and r 0 b(a) da = +∞. This implies that ∀ t, v(t -ε, r -ε, •) = 0 in L 2 (Ω) as ε → 0.
Now, using the inverse transform (34), we obtain (u, v) that satisfy ( 11)-( 14). This proves the existence part of Theorem 3.1. The uniqueness part follows immediately from the uniqueness of the solution of (32) (as given in Theorem 4.1), and the transforms (31) and (34). Hence Theorem 3.1 is proved.

-Proof of Theorem 3.2

The proof of Theorem 3.2 is based on the same idea: consider (u, v) the solution [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF], and (û, v) given by (31) the solution of (32) with the parameters (33). a) First assume that ( 8), ( 9), ( 10) and ( 15) are satisfied. The iterative procedure described below gives that

(37) f0 ≥ 0, û0 ≥ 0, v0 ≥ 0 =⇒ v ≥ 0, û ≥ 0.
Indeed, take h 0 = 0 and solve (36): the weak maximum principle implies that v(0) ≥ 0, and then that û(0) ≥ 0. Then, in a second step, take h 1 = û(0) . Then one again, the weak maximum principle implies that v(h1) ≥ 0, and then that û(h1) ≥ 0. Repeating the procedure, we obtain that û ≥ 0 and v ≥ 0. Now we go back to [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF]: clearly (37) and (31) immediately imply that v ≥ 0, and u -V ≥ 0, which of course imply ( 16) and [START_REF] Pozio | Some conditions for global asymptotic stability of equilibria of integro-differential equations[END_REF].

Remark. We could obtain ( 16) and directly, without the transformation (31), as we explain in the following.

b) Now assume that ( 8), ( 9), ( 10) and ( 18) are satisfied. Then of course we can apply what we have just proved and ( 16) and ( 17) are true. It remains to prove that (19) is satisfied. Consider

w(t, x) := (1 -θ)u(t, x) -V (t, x). Denote V(t, x) := r 0 b(a)v(t, a, x) da.
Then we see that (u, V ) satisfies

       u t -∆u + du = V for t > 0, V t -∆V + dV = -V + f 0 (u -V ) -(d -d)V for t > 0, u(t, x) |∂Ω = 0 = V (t, x) |∂Ω for t > 0, x ∈ ∂Ω, u(0, x) = u 0 (x), V (0, x) = V 0 (x) for x ∈ Ω. Hence w satisfies    w t -∆w + dw = (2 -θ)V -f 0 (u -V ) + (d -d)V for t > 0, w(t, x) |∂Ω = 0 for t > 0, x ∈ ∂Ω, w(0, x) = (1 -θ)u 0 (x) -V 0 (x) for x ∈ Ω. Since w = (1 -θ)(u -V ) -θV,
we have

w t -∆w + (d + f 0 1 1 -θ )w = (2 -θ)V + (d -d -f 0 θ 1 -θ )V,
and the weak maximum principle implies that (19) is satisfied. The proof of Theorem 3.2 is completed.

-Proof of Theorem 3.3

The proof of Theorem 3.2 is based on the same idea: consider (u, v) the solution [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF], and (û, v) given by (31) the solution of (32) with the parameters (33). Then we follow the steps of the proof of Theorem 4.3:

• First we decompose û0 , v0 and û, v in Fourier series: let (ϕ j ) j∈N be the orthonormal basis of L 2 (Ω) composed by the eigenfunctions of the Laplace operator with Dirichlet boundary conditions: where, for each j ≥ 0, (û j , vj ) is solution of the problem without diffusion Of course u = 2û + V, and

(38) -∆ϕ j (x) = λ j ϕ j (x) x ∈ Ω, ϕ j (x) = 0 x ∈ ∂Ω,
             ûj t (t) + ( d + λ j )û j (t, x) = r 0 b(a)v j (t,
u(t, •) 2 L 2 (Ω) = ∞ j=0 2û j (t) + r 0 vj (t, a) da 2 .
Hence we have to study the behavior of ûj and vj with respect to t and j.

• Note that since ϕ 0 is positive in Ω, then

û0 0 := Ω û0 (x)ϕ 0 (x) dx
is nonnegative, and û0 0 = 0 if and only if û0 = 0 in Ω. The same remark holds true for v0 . In the same spirit, since û and v are nonnegative, û0 (t) ≥ 0 and v0 (t, a) ≥ 0.

R0 < d + λ 0 =⇒ α * < d + λ 0 , R0 = d + λ 0 =⇒ α * = d + λ 0 , R0 > d + λ 0 =⇒ α * > d + λ 0 .
• Using classical properties of the Laplace transform (see Theorem 3.3 in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]), the following holds:

(39) û0 (t) ∼ c 0 0 e (α * -d-λ0)t , where c 0 0 > 0 (and is given in (3.17) in [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF]).

• Now we are in position to distinguish the three cases of Theorem 3.3:

-Case (iii): R0 > d + λ 0 . Then, as noted, we have α * > d + λ 0 , and (39) implies that û0 (t) goes exponentially fast to +∞ as t → +∞. Since v0 ≥ 0, and

u(t, •) 2 L 2 (Ω) ≥ 2û 0 (t) + r 0 v0 (t, a) da 2 ,
we obtain what is claimed in Theorem 3.3, (iii).

-Case (i): R0 < d + λ 0 . Then as noted we have α * < d + λ 0 , and (39) implies that û0 (t) goes exponentially fast to 0 as t → +∞. In the same way, all the components ûj go exponentially fast to 0, and more precisely, there exists some C such that

û(t, •) 2 L 2 (Ω) ≤ Ce 2(α * -d-λ0)t ( u 0 2 L 2 (Ω) + v 0 2 L 2 ((0,r)×Ω) ).
Since for t > a, we have vj (t, a) = f0 ûj (t -a)e -( d +λj )a-b(a) , we obtain a similar uniform exponential decay for all the vj , and hence for u(t, •) 2 L 2 (Ω) and v 2 L 2 ((0,r)×Ω) as claimed in Theorem 3.3, (i).

-Case (ii). We decompose û(t, x) = û0 (t)ϕ 0 (x) + ∞ j=1 ûj (t)ϕ j (x), and v(t, a, x) = v0 (t, a)ϕ 0 (x) + ∞ j=1 vj (t, a)ϕ j (x); the previous argument of Case (i) gives that all the components ûj and vj go exponentially fast to 0 uniformly with respect to j ≥ 1, and the first components û0 and v0 converge to positive constants. Hence we obtain Theorem 3.3, (ii).

-Conclusion and perspectives

Finally, we corrected the first model ( 22) introduced in [START_REF] Fragnelli | A population equation with diffusion[END_REF][START_REF] Nickel | Positivity and stability of delay equations with nonautonomous past[END_REF] to obtain the more realistic one [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF]. Now we are ready to turn to some other questions, like for example the introduction of more general models or the study of controllability properties of such models.

-Introduction of more general models

Let us recall the Mc Kendrick-Von Foerster model with diffusion which is a classical population dynamics model studied in particular in [START_REF] Anit ¸a | Analysis and control of age-dependent population dynamics[END_REF]. In this model, the density of population u(A, t, x) of age A at time t and located at space x satisfies the following equations (where another nonlocal birth process arises in a boundary condition) :

(40)          u t + u A -∆u + d(A)u = 0, u(A, t, x) |∂Ω = 0, u(0, t, x) = A+ 0 β(A)u(A, t, x) dA, u(A, 0, x) = u 0 (A, x).
Here A ∈ (0, A + ) denotes the age of the individuals and A + represents the maximal life expectancy of an individual, d(A) is the natural death rate, whereas β(A) denotes the natural fertility rate. In summary, (40) is a population dynamics model with diffusion and with age dependence, i.e. structured in space and age.

On the other hand, in models ( 22) and ( 7) introduced here, we recall that no age difference is allowed whereas the time of gestation is considered. In other words, models ( 22) and ( 7) are models with diffusion and with time-gestation dependence, i.e. structured in space and time-gestation.

A natural question is now to write such a model where age difference will also be considered i.e. where the density of population u(A, t, x) and the density of pregnant individuals v(A, t, a, x) will depend on age A, time t, space x and age of gestation a. At this stage, combining the results and the methods of [START_REF] Anit ¸a | Analysis and control of age-dependent population dynamics[END_REF] and of [START_REF] Fragnelli | Qualitative properties of a population dynamics system describing pregnancy[END_REF], it should be possible to write and to study such a dynamics population model with diffusion and with age and time-gestation dependence, i.e. structured in space, age and time-gestation.

-Study of controllability properties

Controllability of diffusive population dynamics models is a subject that is still mainly open. But there are some recent works concerning the Mc Kendrick-Von Foerster model with diffusion : see e.g. [START_REF] Ainseba | Exact and approximate controllability of the age and space population dynamics structured model[END_REF][START_REF] Ainseba | Local exact controllability of the age-dependent population dynamics with diffusion[END_REF][START_REF] Ainseba | On a population dynamics control problem with age dependence and spatial structure[END_REF][START_REF] Ainseba | Exact controllability of a nonlinear populationdynamics problem[END_REF][START_REF] Anit ¸a | Analysis and control of age-dependent population dynamics[END_REF] for results of approximate, local exact or exact controllability.

We wish now to study the case of diffusive population dynamics models that also take into account the pregnancy. Notice that the study of controllability of such models has a link with the study made in [START_REF] Martinez | Regional null controllability of a linearized Crocco type equation[END_REF]. Indeed, in models ( 22) or [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF], the second equation in v has the same structure than the Crocco equation (with constant coefficients) : it is a degenerate parabolic equation that couples a transport phenomenon with a diffusion phenomenon. Of course, the situation will be complicated here by the fact that we deal now with a system of two equations with nonlocal coupling terms... On the other hand, let us mention that there is a rather large literature on the subject of optimal control problems for models like (40). See for example [START_REF] Anit ¸a | Analysis and control of age-dependent population dynamics[END_REF][START_REF] Anit ¸a | Optimal impulse-control of population dynamics with diffusion[END_REF][START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF] and the references therein. It would also be interesting to study those kinds of problems for models like (22) or [START_REF] Anit ¸a | Optimal control of a nonlinear population dynamics with diffusion[END_REF].

  )e -( b(s)+( d +λ0)s) ds where ∀s ∈ (0, r), b(s) := s 0 b(σ) dσ.

5. 1 -

 1 Connection between the problems (7) and (22) 5.1.1 -From (7) to (22)

  with (λ j ) j∈N ⊂ R * + such that λ j +∞ as j → +∞. If we write û0 (x) = ∞ j=0 ûj 0 ϕ j (x), and v0 (a, x) = ∞ j=0 vj 0 (a)ϕ j (x), then û(t, x) = ∞ j=0 ûj (t)ϕ j (x), and v(t, a, x) = ∞ j=0 vj (t, a)ϕ j (x),

  a) da t > 0, vj t (t, a) + vj a (t, a) + ( d + λ j )v j (t, a) = -b(a)v j (t, a) t > 0, a ∈ (0, r), vj (t, 0) = f0 ûj (t) t > 0, ûj (0) = ûj 0 vj (0, a) = vj 0 (a) a ∈ (0, r).

•

  Define R0 by (21); since the function ψ : α → f0 r 0 b(a)e -b(a) e -( d -d)a e -αa da -α is strictly decreasing, there exists one and only one α * > 0 such that f0 r 0 b(a)e -b(a) e -( d -d)a e -α * a da = α * ; moreover, since ψ( d + λ 0 ) = R0 -( d + λ 0 ), we have that

A c k n o w l e d g m e n t s. The first author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and acknowledges the GNAMPA project 2015 "Analisi e controllo di equazioni a derivate parziali nonlineari" for its financial support.

Genni Fragnelli

Università degli Studi di Bari, Dipartimento di Matematica, Via Orabona, 4, 70125 Bari (Italy) e-mail: genni.fragnelli@uniba.it

Patrick Martinez

University of Toulouse Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse cedex 9 (France) e-mail: martinez@math.univ-toulouse.fr

Judith Vancostenoble

University of Toulouse Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse cedex 9 (France) e-mail: vancoste@math.univ-toulouse.fr