P Cannarsa
email: cannarsa@mat.uniroma2.it

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

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

SHARP ESTIMATE OF THE COST OF CONTROLLABILITY FOR A DEGENERATE PARABOLIC EQUATION WITH INTERIOR DEGENERACY

Keywords: 1991 Mathematics Subject Classification. 35K65, 93B05, 93B60, 33C10, 35P10, 30D15 Controllability, degenerate parabolic equation, biorthogonal family

This work is motivated by the study of null controllability for the typical degenerate parabolic equation with interior degeneracy and one-sided control:

ut

).

The goal of the present work is to provide optimal upper and lower estimates of the null controllability cost, with respect to the degeneracy parameter (when α → 1 -) and in short time (when T → 0 + ). We prove that the null controllability cost behaves as 1 1-α as α → 1 -and as e 1/T as T → 0 + . Our analysis is based on the construction of a suitable family biorthogonal to the sequence (e λnt )n in L 2 (0, T ), under some general gap conditions on the sequence (λn)n, conditions that are suggested by a motivating example.

Introduction

General considerations.

Degenerate parabolic equations have received increasing attention in recent years because of their connections with several applied domains such as climate science, populations genetics, vision, and mathematical finance (see, e.g., [START_REF] Cannarsa | Global Carleman estimates for degenerate parabolic operators with applications[END_REF][START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF] and the references therein). Indeed, in all these fields, one is naturally led to consider parabolic problems where the diffusion coefficients lose uniform ellipticity. Different situations may occur: degeneracy (of uniform ellipticity) may take place at the boundary or in the interior of the space domain. Moreover, the equation may be degenerate on a small set or even on the whole domain.

From the point of view of control theory, interesting phenomena have been pointed out for degenerate parabolic equations, in particular the existence of threshold values, where some property completely changes its nature (for examples, in several examples, null controllability holds below some critical value but not above). We refer the reader to [START_REF] Cannarsa | Carleman estimates for a class of degenerate parabolic operators[END_REF][START_REF] Cannarsa | Global Carleman estimates for degenerate parabolic operators with applications[END_REF] for boundary-degenerate parabolic operators and to [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF][START_REF] Beauchard | Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type[END_REF] for interior-degenerate equations, associated with certain classes of hypoelliptic diffusion operators (and see also [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF]) for Grushin type structures, and to [START_REF] Beauchard | Heat equation on the Heisenberg group: observability and applications[END_REF] for the Heisenberg operator.

1.2. The problem considered here, and the main results.

In [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF], null controllability for the following parabolic equation (1. 1)

         u t -(|x| α u x ) x = h(x, t)χ (a,b) ,
x ∈ (-1, 1) u(-1, t) = 0, t ∈ (0, T ), u(1, t) = 0, t ∈ (0, T ), u(x, 0) = u 0 (x),

x ∈ (-1, 1),

• with interior degeneracy (at point x = 0),

• and using a one-sided control (localized in (a, b) with 0 < a < b < 1), has been studied from both the theoretical and numerical point of view. It turns out that null controllability,

• fails for α ∈ [1, 2),

• holds true when α ∈ [0, 1).

Consequently, the control acting on (a, b) is sufficiently strong to cross the degeneracy point x = 0 if and only if α < 1. A tool to measure the change of behavior for α = 1 is to estimate the "null controllability cost", that is:

• given u 0 , consider the set of admissible controls h driving the solution of (1. 1) to rest in time T :

U ad (α, T ; u 0 ) := h ∈ L 2 ((a, b) × (0, T )) | u (h) (T ) = 0 .
• then, given u 0 , consider the norm of the best admissible control: inf h∈U ad (α,T ;u0) h L 2 ((a,b)×(0,T )) ,

and maximize this quantity along u 0 in the unit ball or sphere of L 2 (-1, 1), hence, roughly speaking, the smaller quantity that one needs to control all the initial conditions of the unit ball or sphere of L 2 (-1, 1):

(1. 2) C N C (α, T ) := sup u0 L 2 (-1,1) =1 inf h∈U ad (α,T ;u0) h L 2 ((a,b)×(0,T )) ,

• finally, estimate the behavior of the null controllability cost C N C (α, T ) as α → 1 -.

It was proved in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF] that there exist constants C, C > 0, independent of α ∈ [0, 1) and of T > 0, such that 2 e C/T e -T /C . Therefore:

(1. 3) C (1 -α) √ T e -T /C ≤ C N C (α, T ) ≤ C (1 -α)
• C N C (α, T ) blows up as α → 1 -(as expected since null controllability does not hold for α = 1), with a blow-up rate between 1 1-α and 1

(1-α) 2 , • and C N C (α, T ) blows up as T → 0 + , with a blow-up rate between 1 √ T and e 1/T . The goal of the present paper is to improve the estimates (1. 3), proving that • C N C (α, T ) blows up exactly as 1 1-α when α → 1 -, • and C N C (α, T ) blows up exponentially as e 1/T when T → 0 + . (See Theorems 2.1 and 2.2 for a precise statement.) 1.3. Main tools and comparison with the literature.

Degenerate parabolic equations with one (or more) degeneracy point inside the domain have also been studied

• by the flatness method developed by Martin-Rosier-Rouchon in [START_REF] Martin | Null controllability of the 1D heat equation using flatness[END_REF][START_REF] Martin | Null controllability of the heat equation using flatness[END_REF][START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] (see also Moyano [35] for some strongly degenerate equations), • by Carleman estimates, see Fragnelli-Mugnai in [START_REF] Fragnelli | Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations[END_REF][START_REF] Fragnelli | Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients[END_REF] when the control region is on both sides of the space domain with respect to the degeneracy point.

However, our analysis of the cost in the weakly degenerate case does not seem to be attainable by these approaches. It is based on the spectral problem associated with (1. 1). The eigenvalues of problem (1. 1) are related to the zeros of Bessel functions, and exhibit the following behavior: the gap between an eigenvalue of odd order and the consecutive one goes to 0 as α → 1 -(see [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF])

∀n ≥ 1, λ 2n (α) -λ 2n-1 (α) → 0 as α → 1 -,
while there is a uniform gap for the other ones: there is some

C u > 0 independent of α ∈ [0, 1) and of n ≥ 1 such that ∀α ∈ [0, 1), ∀n ≥ 1, λ 2n+1 (α) -λ 2n (α) ≥ C u .
Let us observe that recently Benabdallah-Boyer-Morancey [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF] provided general results concerning such problems where groups of eigenvalues are separated by a uniform gap but, inside each group, eigenvalues can be close. This motivated us to push further the techniques we developed earlier, designed to study precisely the effects of some parameters (α and T here) on the null controllability cost, whereas [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF] is focused on obtaining formulas giving the minimal time needed to control a given system. Our approach is the following:

• to obtain a precise upper estimate of the null controllability cost:

first we prove a general result (Theorem 2.3) concerning biorthogonal families to exponentials in the case where pairs of eigenvalues are close; the proof is based on complex analysis, and in the spirit of a similar result proved in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF] (but for which we needed to modify the starting point of the proof), and then Theorem 2.3 allows us to deduce that C N C (α, T ) blows up at most as 1 1-α ; • to obtain an exponential lower estimate of the null controllability cost:

we add an artificial control region, the goal being to deal with a new eigenvalue problem that is easy to study, and that will of course make the related null controllability cost cheaper than the original one, and then we estimate the new controllability cost with some Hilbertian techniques developed also in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], in the spirit of a result of Guichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF], and we conclude.

1.4. Plan of the paper.

• In section 2, we state our results: -Theorem 2.1 (upper estimate), -Theorem 2.2 (lower estimate), and Theorem 2.3 (general construction of a biorthogonal family under this assumption that pairs of eigenvalues condensate). • In section 3, we prove Theorem 2.3.

• In section 4, we prove Theorem 2.1.

• In section 5, we prove Theorem 2.2.

Main results

Upper bound of the null controllability cost.

Our first result is the following upper estimate of the null controllability cost, more precise than the one in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF]: Theorem 2.1. There exists C u > 0, independent of α ∈ [0, 1) and T > 0, such that

(2. 1) ∀α ∈ [0, 1), ∀T > 0, C N C (α, T ) ≤ C u 1 -α e Cu T e -T Cu .
The proof of Theorem 2.1 is based on a general result, stated in section 2.3.

Lower bound of the null controllability cost.

The following result improves also the lower estimate of the null controllability cost obtained in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF], yielding the expected exponential behavior in short time: Theorem 2.2. There exists c u > 0, independent of α ∈ [0, 1) and of T ∈ (0, 1), such that

(2. 2) ∀α ∈ [0, 1), ∀T ∈ (0, 1), C N C (α, T ) ≥ c u 1 -α , and
(2. 3) ∀α ∈ [0, 1), ∀T ∈ (0, 1), C N C (α, T ) ≥ c u e cu T .
Remark 2.1.

• Note that, comparing to (2. 1), it would have been natural to expect a lower bound of the form

(2. 4) ∀α ∈ [0, 1), ∀T ∈ (0, 1), C N C (α, T ) ≥ c u 1 -α e cu T ,
which does not follow from our results. However, combining (2. 2) and (2. 3), we have

∀α ∈ [0, 1), ∀T ∈ (0, 1), ∀θ ∈ [0, 1], C N C (α, T ) ≥ c u (1 -α) θ e cu (1-θ) T
, which combines the blow-up in α whith the one in T . • Combining with (1. 3), one has that there exists c u independent of α ∈ [0, 1) and of T > 0 such that

∀α ∈ [0, 1), ∀T ∈ (0, 1), C N C (α, T ) ≥ c u e cu T e -T cu ,
and, once again,

∀α ∈ [0, 1), ∀T > 0, ∀θ ∈ [0, 1], C N C (α, T ) ≥ c u (1 -α) θ e cu(1-θ) T e -T cu .

An adapted biorthogonal family.

The proof of Theorem 2.1 is based on the following general result, which can be viewed as a particular version of Theorem 2.3 of [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], but that gives a more precise estimate, that will be useful in our present context: Theorem 2.3. Suppose λ 1 ≥ 0 and assume that there exists 0 < γ min ≤ γ * min such that

(2. 5) ∀m ≥ 1, √ λ 2m -λ 2m-1 ≥ γ min , ∀m ≥ 1, λ 2m+1 - √ λ 2m ≥ γ * min ,
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. 6) ∀m, n ≥ 1, T 0 σ + m (t)e λnt dt = δ m,n .
Moreover, there is a universal constant C u > 0, independent of T , γ min , γ * min and m, such that, for all m ≥ 1, we have

(2. 7) σ + m 2 L 2 (0,T ) ≤ C u 1+ (γ min + γ * min ) 2 γ min ( √ λ 1 + γ min
B(T, γ min , γ * min ) =    1 T + 1 T 2 (γmin+γ * min ) 2 if T ≤ 1 (γmin+γ * min ) 2 , C u (γ min + γ * min ) 2 if T ≥ 1 (γmin+γ * min ) 2 .
(We note that Theorem 2.3 is similar to the results we proved in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF][START_REF] Cannarsa | The cost of controlling degenerate parabolic equations by locally distributed controls[END_REF]. Each of these results has been adapted to different examples and applications. Assumptions change from a version to another, leading to precise results in the desired case. It seems difficult, however, to provide a general framework for this theory. We adapt the strategy developed in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF][START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF] to our assumptions. This approach is adapted from the construction of Seidman-Avdonin-Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], which has the advantage to be completely explicit, 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 quite optimal results.

Our approach is a perturbation of the one used in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF][START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF], based on 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 |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.
In [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF][START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF], we adapted the general construction of [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF] (see Theorem 2 and Lemma 3 in [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF]) to construct a suitable sequence (f m ) m satisfying (3. 1)

     ∀m, n ≥ 1, f m (-iλ n ) = δ m,n , ∀z ∈ C, |f m (-z)e -iz T 2 | ≤ C m e T 2 |z| , ∀m ≥ 1, f m ∈ L 2 (R).
Then the two last properties together with the Paley-Wiener theorem 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 m,n , 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
is 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 m,n , 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. With respect to [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF][START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF], we choose a more adapted infinite product, and then we perform all the necessary estimates.

The counting function.

Consider

∀ρ > 0, N m (ρ) := card {k, 0 < |λ m -λ k | ≤ ρ}.
We prove the following:

Lemma 3.1. Assume that the gap assumptions (2. 5) are satisfied. Then, for all m ≥ 1, we have

(3. 2) ρ ∈ (0, γ min (γ min + 2 λ 1 )) =⇒ N m (ρ) = 0, and
(3. 3) ∀m ≥ 1, ∀ρ > 0, N m (ρ) ≤ 4 √ ρ γ min + γ * min + 1.
Proof of Lemma 3.1. Of course the "worst" situation is when we have equalities in (2. 5), hence when

(3. 4) ∀m ≥ 1, √ λ 2m -λ 2m-1 = γ min , ∀m ≥ 1, λ 2m+1 - √ λ 2m = γ *
min , "worst" in the sense that if (3. 2) and (3. 3) are true under (3. 4), they will be true under (2. 5).

Hence we assume (3. 4). And then, we have easily the following formulas:

(3. 5) ∀m ≥ 1, √ λ 2m = √ λ 1 + mγ min + (m -1)γ * min , ∀m ≥ 1, λ 2m+1 = √ λ 1 + mγ min + mγ * min ,
and then, we obtain clearly that for all m ≥ 1

λ 2m -λ 2m-1 = ( λ 2m -λ 2m-1 )( λ 2m + λ 2m-1 ) = γ min (2 λ 1 + (2m -1)γ min + 2(m -1)γ * min ) ≥ γ min (2 λ 1 + γ min ), and
λ 2m+1 -λ 2m = ( λ 2m+1 -λ 2m )( λ 2m+1 + λ 2m ) = γ * min (2 λ 1 + 2mγ min + (2m -1)γ * min ) ≥ γ * min (2 λ 1 + γ min ) ≥ γ min (2 λ 1 + γ min ). Now, if N m (ρ) = 0, there exists some k such that |λ k -λ m | ≤ ρ,
and of course the same holds true with |k -m| = 1. Then, m is even and k is odd, or the contrary, and there exists some n such that

λ 2n -λ 2n-1 ≤ ρ, or λ 2n+1 -λ 2n ≤ ρ.
In any case, γ min (2

λ 1 + γ min ) ≤ ρ,
which implies (3. 2). Now we prove (3. 3). We distinguish several cases. First we consider that m is even: m = 2m , and we estimate the number of k such that

|λ k -λ 2m | ≤ ρ. • if k > 2m and k is even, hence if k = 2k with k > m , then λ 2k ≤ ρ + λ 2m , hence λ 2k ≤ ρ + λ 2m ≤ √ ρ + λ 2m ;
and using (3. 5), we obtain

λ 1 + k γ min + (k -1)γ * min ≤ √ ρ + λ 1 + m γ min + (m -1)γ * min , which gives (k -m )(γ min + γ * min ) ≤ √ ρ, hence 1 ≤ k -m ≤ √ ρ γ min + γ * min ,
and there are at most

λ 2k +1 ≤ ρ + λ 2m ≤ √ ρ + λ 2m ,
and using (3. 5), we obtain

λ 1 + k γ min + k γ * min ≤ √ ρ + λ 1 + m γ min + (m -1)γ * min , hence (k -m )(γ min + γ * min ) ≤ √ ρ -γ * min , hence 0 ≤ k -m ≤ √ ρ -γ * min γ min + γ * min ,
and there are at most

√ ρ-γ * min γmin+γ * min + 1 such integers k ; • finally, if k < 2m and k is odd, hence k = 2k + 1 with k < m , we have λ 2m ≤ √ ρ + λ 2k +1 ,
and using (3. 5), we obtain

λ 1 + m γ min + (m -1)γ * min ≤ √ ρ + λ 1 + k γ min + k γ * min , hence (m -k )(γ min + γ * min ) ≤ √ ρ + γ * min , hence 0 < m -k ≤ √ ρ + γ * min γ min + γ * min ,
and there are at most

√ ρ+γ * min γmin+γ * min such integers k .
Finally we obtain that

N m (ρ) ≤ √ ρ γ min + γ * min + √ ρ γ min + γ * min + √ ρ -γ * min γ min + γ * min + 1 + √ ρ + γ * min γ min + γ * min = 4 √ ρ γ min + γ * min + 1,
which proves (3. 3).

A Weierstrass product.

Motivated by [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF], we used in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF][START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF] the following product

∞ k=0,k =m 1 - iz -λ m λ k -λ m 2 ,
but under our assumptions it seems more clever to consider the following one:

(3. 6) ∞ k=1,k =m 1 - iz -λ m λ k -λ m =: F m (z). Since ln 1 - iz -λ m λ k -λ m ≤ ln 1 + |z| + λ m |λ k -λ m | ∼ k→∞ |z| + λ m λ k ,
and since k 1 λ k is convergent, we deduce that the infinite product defining F m converges uniformly over all the compacts 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 = 0 if m = n, 1 if m = n, , hence (3. 7) ∀m, n ≥ 1, F m (-iλ n ) = δ m,n .
Now we are going to estimate the growth of F m . We prove the following Lemma 3.2. Assume that the gap assumption (2. 5) is satisfied. Then the function F m satisfies the following growth estimate:

(3. 8) ∀m ≥ 1, ∀z ∈ C, |F m (z)| ≤ 1+ |z| + λ m γ min (2 √ λ 1 + γ min ) e 4π γ min +γ * min ( √ |z|+λm) .
The proof of Lemma 3.2 is based on the following preliminary estimate, relying the growth of F m with the counting function N m : Lemma 3.3. Assume that the gap assumption (2. 5) is satisfied. Then the function F m satisfies the following growth estimate:

(3. 9) ∀m ≥ 1, ∀z ∈ C, ln |F m (z -iλ m )| ≤ +∞ 0 N m (ρ) |z| ρ 2 + ρ|z| dρ.
Proof of Lemma 3.2, assuming Lemma 3.3. Assume that (3. 9) is true. Then using (3. 2) and (3. 3), we obtain that

ln |F m (z -iλ m )| ≤ +∞ γmin(γmin+2 √ λ1) 4 √ ρ γ min + γ * min + 1 |z| ρ 2 + ρ|z| dρ ≤ 4 γ min + γ * min +∞ 0 |z| √ ρ ρ 2 + ρ|z| dρ + +∞ γmin(γmin+2 √ λ1) |z| ρ 2 + ρ|z| dρ,
and these two integrals can be easily computed: for the last one, we have

|z| ρ 2 + ρ|z| = - d dρ ln 1 + |z| ρ
(this will appear in the proof of Lemma 3.3), and then

+∞ γmin(γmin+2 √ λ1) |z| ρ 2 + ρ|z| dρ = -ln 1 + |z| ρ +∞ γmin(γmin+2 √ λ1) = ln 1 + |z| γ min (γ min + 2 √ λ 1 ) ;
for the other one, we first note that

|z| √ ρ ρ 2 + ρ|z| = 1 ρ - 1 ρ + |z| √ ρ = 1 √ ρ - √ ρ ρ + |z| ;
fix X > 0, then we have (using the change of variables σ = √ ρ)

X 0 |z| √ ρ ρ 2 + ρ|z| dρ = X 0 1 √ ρ dρ - X 0 √ ρ ρ + |z| dρ = 2 √ X - √ X 0 σ σ 2 + |z| 2σ dσ = 2 √ X - √ X 0 2σ 2 + 2|z| -2|z| σ 2 + |z| dσ = 2 √ X - √ X 0 2 - 2|z| σ 2 + |z| dσ = √ X 0 2|z| σ 2 + |z| dσ = √ X 0 2|z| |z| 1 1 + σ √ |z| 2 dσ = 2 |z| arctan σ |z| √ X 0 = 2 |z| arctan √ X |z| ,
and letting X → +∞, we obtain that

+∞ 0 |z| √ ρ ρ 2 + ρ|z| dρ = π |z|. Therefore ln |F m (z -iλ m )| ≤ ln 1 + |z| γ min (γ min + 2 √ λ 1 ) + 4π γ min + γ * min |z|, hence |F m (z -iλ m )| ≤ 1 + |z| γ min (γ min + 2 √ λ 1 ) e 4π γ
(z -iλ m )| = ∞ k=1,k =m ln 1 - iz λ k -λ m ≤ ∞ k=1,k =m ln 1 + |z| |λ k -λ m | ,
and the proof of Lemma 3.3 will follow from the following identity:

(3. 10) ∞ k=1,k =m ln 1 + |z| |λ k -λ m | = +∞ 0 N m (ρ) |z| ρ 2 + ρ|z| dρ.
Hence it remains to prove (3. 10). Let us prove it first when m = 1: it comes from the definition of N 1 (ρ) that

N 1 (ρ) = card {k > 1, λ k -λ 1 ≤ ρ}, hence 0 ≤ ρ < λ 2 -λ 1 =⇒ N 1 (ρ) = 0, λ 2 -λ 1 ≤ ρ < λ 3 -λ 1 =⇒ N 1 (ρ) = 1, λ 3 -λ 1 ≤ ρ < λ 4 -λ 1 =⇒ N 1 (ρ) = 2,
and more generally

∀k ≥ 1 : λ k -λ 1 ≤ ρ < λ k+1 -λ 1 =⇒ N 1 (ρ) = k -1.
Then, given N ≥ 1, we have

λ N +1 -λ1 0 N 1 (ρ) |z| ρ 2 + ρ|z| dρ = N k=1 λ k+1 -λ1 λ k -λ1 N 1 (ρ) |z| ρ 2 + ρ|z| dρ = N k=1 λ k+1 -λ1 λ k -λ1 (k -1) |z| ρ 2 + ρ|z| dρ = N k=2 (k -1) λ k+1 -λ1 λ k -λ1 - d dρ ln 1 + |z| ρ dρ = N k=2 (k -1) -ln 1 + |z| ρ λ k+1 -λ1 λ k -λ1 = N k=2 (k -1) ln 1 + |z| λ k -λ 1 -ln 1 + |z| λ k+1 -λ 1 = N k=2 (k -1) ln 1 + |z| λ k -λ 1 - N k=2 (k -1) ln 1 + |z| λ k+1 -λ 1 = N k=2 (k -1) ln 1 + |z| λ k -λ 1 - N +1 k=3 (k -2) ln 1 + |z| λ k -λ 1 = N k=2 ln 1 + |z| λ k -λ 1 -(N -1) ln 1 + |z| λ N +1 -λ 1 .
To conclude, let N → ∞:

ln 1 + |z| λ N +1 -λ 1 ∼ N →∞ |z| λ N +1 -λ 1 ∼ N →∞ |z| λ N +1 , hence (N -1) ln 1 + |z| λ N +1 -λ 1 ∼ N →∞ |z| N λ N +1 → 0 as N → ∞
since there is c > 0 such that λ N ≥ mN 2 . Therefore we obtain

+∞ 0 N 1 (ρ) |z| ρ 2 + ρ|z| dρ = ∞ k=2 ln 1 + |z| λ k -λ 1 , hence (3. 10) in the case m = 1.
The case m = 2 can be studied in a similar way, or performing the following change: denote λ1 the symmetric of λ 1 with respect to λ 2 : λ1 -

λ 2 = λ 2 -λ 1 ,
and reorder the sequence {λ n , n ≥ 2} ∪ { λ1 } in the increasing order: then we are in the situation of the previous case, and we obtain that

+∞ 0 N 2 (ρ) |z| ρ 2 + ρ|z| dρ = ln 1 + |z| λ1 -λ 2 + ∞ k=3 ln 1 + |z| λ k -λ 2 , = ∞ k=1,k =2 ln 1 + |z| |λ k -λ 2 | ,
which is (3. 10) when m = 2. And the same reasoning allows to prove (3. 10) in full generality. This concludes the proof of Lemma 3.3.

The direct consequences.

We derive from Lemma 3.2 that, for all m ≥ 1, for all z ∈ C, we have

|F m (z)| ≤ 1 + |z| + λ m (γ min + γ * min ) 2 (γ min + γ * min ) 2 γ min (2 √ λ 1 + γ min ) e 4π γ min +γ * min ( √ |z|+λm) ,
hence there exists some C u > 0 independent of m, γ min , γ * min and z such that

(3. 11) |F m (z)| ≤ C u 1 + (γ min + γ * min ) 2 γ min (2 √ λ 1 + γ min ) e 8π γ min +γ * min ( √ |z|+ √ λm) .
The main differences with respect to the general result in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], under our new assumptions (2. 5), are • the coefficient in the exponential, depending on 1/(γ min + γ * min ) instead of 1/γ min ,

• the multiplicative coefficient 1 + (γmin+γ * min ) 2 γmin(2 √
λ1+γmin) : since we have in mind examples where γ min is small and γ * min is not small, the coefficient in the exponential will be of the order 1/γ * min (hence not large), and the only large coefficient is the multiplicative one, of the order 1/γ min if λ 1 > 0.

Then we can proceed as in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], taking into account these changes:

• We choose (3. 12) T := min{T, 1 (γ min + γ * min ) 2 }, and

(3. 13) N ≥ 2 + θ 3 (γ min + γ * min )T
with a suitable θ 3 (independent of T > 0 and of m ≥ 0, and given in (3. 18)).

• We consider 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 the associated mollifier

     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. 16)      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. 17) ∀x ∈ R + , P N ,T (ix) ≥ e -θ2 √ C N ,T x .
Using these parameters, we define

(3. 18) θ 3 = 2 11 θ 0 π 2 θ 2 1
.

• Finally we consider

(3. 19) ∀m ≥ 1, ∀z ∈ C, f m,N ,T (z) := F m (z) P N ,T (-z) P N ,T (iλ m ) ,
and we have the following Lemma 3.5. When T and N satisfy (3. 12) and (3. 13), the functions f m,N ,T are entire and satisfy the following properties:

for all m, n ≥ 1, we have

(3. 20) f m,N ,T (-iλ n ) = δ m,n ;
for all m ≥ 1, for all ε > 0, there exists C m,γmin,γ * min ,N ,T ,ε > 0 such that

(3. 21) ∀z ∈ C, |f m,N ,T (-z)e -iz T 2 | ≤ C m,γmin,γ * min ,N ,T ,ε e ( T 2 +ε)|z| ; -for all m ≥ 1, f m,N ,T ∈ L 2 (R).
(The proof of Lemma 3.5 is directly adapted from the one of Lemma 4.4 of [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], taking into account the new estimate (3. 11).

3.5. End of the proof of Theorem 2.3: 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 ( [START_REF] Young | An Introduction to Nonharmonic Fourier Series[END_REF] p. 100) shows that

φ m,N ,T is compactly supported in [-T 2 -ε, T 2 + ε] (thanks to (3. 21)). 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. 22) 2 + θ 3 (γ min + γ min ) 2 T ≤ N ≤ 4 + θ 3 (γ min + γ min ) 2 T .
Then we have the following Lemma 3.6. Take T and N satisfying (3. 12) and (3. 22), and consider

(3. 23) σ + m,N ,T (t) := φ m,N ,T ( T 2 -t)e -λmT .
Then the family (σ + m,N ,T ) m≥0 is biorthogonal to the family (e λnt ) n≥0 in L 2 (0, T ):

(3. 24) ∀m, n ≥ 1, T 0 σ + m,N ,T (t)e λnt dt = δ m,n .
Moreover, it satisfies: there is some universal constant C u independent of T , γ * min , and m such that, for all m ≥ 1, we have where B(T, γ min , γ * min ) is given by (2. 8). Proof of Lemma 3.6. It is similar to the proof of Lemma 4.5 of [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], taking into account (3. 11).

Proof of Theorem 2.1

First we recall from [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF] some useful properties, that we will need to use.

Useful properties ([7]

): well-posedness, eigenvalues and eigenfunctions.

4.1.a. Functional setting and well-posedness ([7]).

For 0 ≤ α < 1, we consider

(4. 1) H 1 α (-1, 1) := u ∈ L 2 (-1, 1) | u absolutely continuous in [-1, 1], 1 -1 |x| α u 2
x dx < ∞ and u(-1) = 0 = u(1) .

H 1 α (-1, 1) is endowed with the natural scalar product

∀f, g ∈ H 1 α (-1, 1), (f, g) = 1 -1 |x| α f (x)g (x) + f (x)g(x) dx.
Next, consider Given α ∈ [0, 1), we have the following: a)

H 2 α (-1, 1) := u ∈ H 1 α (-1, 1) | 1 -1 |(|x| α u (x)) | 2 dx < ∞ ,
H 1 α (-1, 1) is a Hilbert space; b) A : D(A) ⊂ L 2 (-1, 1) → L 2 (-1, 1
) is a self-adjoint negative operator with dense domain.

Hence, A is the infinitesimal generator of an analytic semigroup of contractions e tA on L 2 (-1, 1). Given a source term h in L 2 ((-1, 1) × (0, T )) and an initial condition v 0 ∈ L 2 (-1, 1), consider the problem (4. 2)

     v t -(|x| α v x ) x = h(x, t), v(-1, t) = 0 = v(1, t), v(x, 0) = v 0 (x). The function v ∈ C 0 ([0, T ]; L 2 (-1, 1)) ∩ L 2 (0, T ; H 1 α (-1, 1
)) given by the variation of constant formula v(•, t) = e tA v 0 + t 0 e (t-s)A h(•, s) ds is called the mild solution of (4. 2). We say that a function

v ∈ C 0 ([0, T ]; H 1 α (-1, 1)) ∩ H 1 (0, T ; L 2 (-1, 1)) ∩ L 2 (0, T ; D(A)) is a strict solution of (4. 2) if v satisfies v t -(|x| α v x ) x = h(x, t
) almost everywhere in (-1, 1)×(0, T ), and the initial and boundary conditions are fulfilled for all t ∈ [0, T ] and all x ∈ [-1, 1]. And we have the following

Proposition 4.2. ([7]) If v 0 ∈ H 1 α (-1, 1)
, then the mild solution of (4. 2) is the unique strict solution of (4. 2).

4.1.b. Eigenvalues and eigenfunctions ([7]).

Now we recall from [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF] the eigenvalues and associated eigenfunctions of the degenerate diffusion operator u → -(|x| α u ) , i.e. the solutions (λ, Φ) of

(4. 3) -(|x| α Φ (x)) = λΦ(x) x ∈ (-1, 1), Φ(-1) = 0 = Φ(1).
Let us recall some notations:

• when α ∈ [0, 1), let

ν α := |α -1| 2 -α = 1 -α 2 -α , κ α := 2 -α 2 ,
• and given ν > 0, we also denote J ν the Bessel function of positive order ν, J -ν the Bessel function of negative order -ν, (j ν,m ) m≥1 the sequence of positive zeros of J ν and (j -ν,m ) m≥1 the sequence of positive zeros of J -ν (see of course Watson [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], Lebedev [START_REF] Lebedev | Special Functions and their Applications[END_REF], and the useful properties [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF]). Then we have the following description for (4. 3), see Proposition 2.7 and also equation ( 25) in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF]: when α ∈ [0, 1), we have exactly two sub-families of eigenvalues and associated eigenfunctions for problem (4. 3), that is:

• the eigenvalues of the form κ 2 α j 2 να,n , associated with the odd functions

(4. 4) Φ (o) α,n (x) = x 1-α 2 J να (j να,n x κα ) if x ∈ (0, 1) -|x| 1-α 2 J να (j να,n |x| κα ) if x ∈ (-1, 0) ,
• the eigenvalues of the form κ 2 α j 2 -να,n , associated with the even functions

(4. 5) Φ (e) α,n (x) = x 1-α 2 J -να (j -να,n x κα ) if x ∈ (0, 1) |x| 1-α 2 J -να (j -να,n |x| κα ) if x ∈ (-1, 0) .
Moreover, the family {Φ

(o) α,n , Φ (e)
α,n , n ≥ 1} forms an orthogonal basis of L 2 (-1, 1). It is easy and practical to order the eigenvalues: since J να (0) = 0 and the zeros of J να and J -να are interlaced (because of Sturm's theorems), we have

0 < j -να,1 < j να,1 < j -να,2 < j να,2 < • • • , hence it is natural to denote (4. 6) ∀n ≥ 1, λ α,2n-1 := κ 2 α j 2 -να,n , λ α,2n := κ 2 α j 2 να,n , hence in such a way that 0 < λ α,1 < λ α,2 < λ α,3 < λ α,4 < • • • ,
and the associated normalized eigenfunctions

(4. 7) ∀n ≥ 1, Φα,2n-1 := √ κ α |J -να (j -να,n )| Φ (e) α,n , Φα,2n := √ κ α |J να (j να,n )| Φ (o) α,n
form an orthonormal basis of L 2 (-1, 1).

Upper estimate of the cost of controllability.

We use the moment method. First we expand the initial condition u 0 ∈ L 2 (-1, 1): there exists (µ 0 α,n ) n≥1 ∈ 2 (N ) such that

u 0 (x) = n≥1 µ 0 α,n Φα,n (x),
x ∈ (-1, 1), and we see that h is a control that drives the solution of (1. 1) to 0 in time T if and only if

(4. 8) ∀n ≥ 1, T 0 1 -1 h(x, t)χ [a,b] (x) Φα,n (x)e λα,nt dxdt = -µ 0 α,n .
And if the sequence (σ + α,m ) m≥1 is biorthogonal to the sequence (e λα,nt ) n≥1 in L 2 (0, T ) and satisfies suitable upper estimates, then the function defined by

(4. 9) h(x, t) := m≥1 -µ 0 α,m σ + α,m (t) Φα,m (x) b a Φ2 α,m
belongs to L 2 ((-1, 1) × (0, T )) and satisfies the moment problem (4. 8) (see, e.g. [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF][START_REF] Cannarsa | The cost of controlling degenerate parabolic equations by locally distributed controls[END_REF]). We would like to use Theorem 2.3, so we need to check that (2. 5) holds in our case. We prove the following: Lemma 4.1. There exist m 1 , m 2 > 0 independent of α ∈ [0, 1) and there exists α * ∈ [0, 1) such that

(4. 10) ∀α ∈ [α * , 1), ∀n ≥ 1, λ α,2n -λ α,2n-1 ≥ m 1 (1 -α) =: γ min (α), and
(4. 11) ∀α ∈ [α * , 1), ∀n ≥ 1, λ α,2n+1 -λ α,2n ≥ m 2 =: γ * min (α).
Proof of Lemma 4.1. This follows from Lemma 6.3 in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF], and its consequence: using an integral formula:

dj ν,n dν = 2j ν,n +∞ 0 K 0 (2j ν,n sinh t)e -2νt dt,
(see Watson [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], p. 508), it is proved that there exist m * , M * and α * such that 0 < m * < M * , α * ∈ (0, 1) and such that

∀α ∈ [α * , 1), ∀n ≥ 1, m * (1 -α) ≤ j να,n -j -να,n ≤ M * (1 -α),
and as a consequence

∀α ∈ [α * , 1), ∀n ≥ 1, m * κ α (1 -α) ≤ λ α,2n -λ α,2n-1 ≤ M * κ α (1 -α).
The left part already gives (4. 10). For the right part, it is sufficient to note that the gap λ α,2n -λ α,2n-1 (uniformly small, of the order 1 -α) allows us to estimate the gap λ α,2n+1 -λ α,2n : indeed,

λ α,2n+1 -λ α,2n = λ α,2n+1 -λ α,2n-1 - λ α,2n -λ α,2n-1 ≥ κ α j -να,n+1 -j -να,n -M * (1 -α);
from Komornik-Loreti [START_REF] Komornik | Fourier Series in Control Theory[END_REF] p. 135, the sequence (j -να,n+1 -j -να,n ) n≥1 is nondecreasing, and converges to π, hence

j -να,n+1 -j -να,n ≥ j -να,2 -j -να,1 ,
and since the function α → j -να,2 -j -να,1 is continuous on [0, 1), positive and has a positive limit (equal to j 0,2 -j 0,1 ) when α → 1 -, there exists m 0 > 0 such that ∀α ∈ [0, 1), ∀n ≥ 1, j -να,n+1 -j -να,n ≥ j -να,2 -j -να,1 ≥ m 0 , and then

λ α,2n+1 -λ α,2n ≥ 1 2 m 0 -M * (1 -α),
which completes the proof of Lemma 4.1 (taking eventually another α * closer to 1). Now, Lemma 4.1 gives that the condition (2. 5) is satisfied (with explicit values of γ min and γ * min ), then Theorem 2.3 applies, and we can use the sequence (σ + α,m ) m≥1 constructed in Theorem 2.3: using (2. 7), and the estimate (47) in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF]:

∃m * , ∀α ∈ [0, 1), ∀n ≥ 1, b a Φ2 α,m ≥ m * ,
in (4. 9), we obtain From classical estimates, see for example Lorch-Muldoon [START_REF] Lorch | Monotonic sequences related to zeros of Bessel functions[END_REF]:

h 2 L 2 ((-1,1)×(0,T )) = m≥1 |µ 0 α,m | 2 σ + α,m 2 L 2 (0,T ) 1 b a Φ2 α,m 2 ≤ m≥1 |µ 0 α,m | 2 sup m≥1 σ + α,m 2 L 2 (0,T ) 1 b a Φ2 α,m 2 ≤ u 0 2 L 2 (-1,1) C u m * 2 1 + (γ min (α) + γ * min (α)) 2 γ min (α)( λ α,1 + γ min (α))
(4. 12) ∀ν ∈ [0, 1 2 ], ∀n ≥ 1, π(n + ν 2 - 1 4 ) ≤ j ν,n ≤ π(n + ν 4 - 1 8
),

there exists 0 < C 1 < C 2 independent of α ∈ [0, 1) such that ∀α ∈ [0, 1), ∀m ≥ 1, C 1 m 2 ≤ λ α,m ≤ C 2 m 2 ;
hence we have

∀α ∈ [0, 1), ∀m ≥ 1, e Cu √ λα,m γ min (α)+γ * min (α) e -2λα,mT ≤ e Cu √ C2m-2C1T m 2
, and studying the function

y ∈ [0, +∞) → C u √ C 2 y -2C 1 T y 2 , we see that e Cu √ C2m-2C1T m 2 ≤ e C3/T with C 3 = C 2 u C 2 8C 1 .
Finally, since α → λ α,1 is continuous on [0, 1), positive and with a positive limit ( 1 4 j 2 0,1 ) as α → 1 -, there exists some m * such that

λ α,1 ≥ m * .
Hence

h 2 L 2 ((-1,1)×(0,T )) ≤ u 0 2 L 2 (-1,1) C u m * 2 1 + C 4 γ min (α) 2 e (C3+ Cu m 2 2 ) 1 T B(T, γ min (α), γ * min (α)), hence C N C (α, T ) ≤ √ C u m * 1 + C 4 γ min (α) e (C3+ Cu m 2 2 ) 1 2T B(T, γ min (α), γ * min (α)).
This gives the expected behavior of the null controllability cost and completes the proof of Theorem 2.1.

5.

Proof of Theorem 2.2 5.1. The initial remark given by the moment method.

The initial remark goes back to the relation given by the moment method: looking to (4. 8), we see that if (1. 1) is null-controllable in time T , then choosing u 0 := -Φ α,1 , any control h 1 that drives the initial condition -Φ α,1 to 0 in time T satisfies

(5. 1) ∀n ≥ 1, T 0 b a h 1 (x, t) Φα,n (x)e λα,nt dxdt = δ n,1 .
Let us take now u 0 := -Φ α,2 , any control h 2 that drives the initial condition -Φ α,2 to 0 in time T satisfies

(5. 2) ∀n ≥ 1, T 0 b a h 2 (x, t) Φα,n (x)e λα,nt dxdt = δ n,2 .
And then, if (1. 1) is null-controllable in time T , choose any control h m that drives the initial condition -Φ α,m to 0 in time T , and the related sequence (h m ) m≥1 is then biorthogonal to the sequence (Φ α,n (x) e λα,nt ) n≥1 in the space L 2 ((a, b) × (0, T )).

Hence it is natural to study biorthogonal sequences to the sequence (Φ α,n (x) e λα,nt ) n≥1 in the space L 2 ((a, b) × (0, T )). We claim the following:

Lemma 5.1. E(α, T, a, b) is a proper subspace of L 2 ((a, b) × (0, T )). Proof of Lemma 5.1. If E(α, T, a, b) = L 2 ((a, b) × (0, T )), then any element of L 2 ((a, b) × (0, T ))
is the limit of a sequence of linear combinations of εα,n . In particular, choose f ∈ L 2 (0, T ), then f is the limit of such a sequence, and integrating with respect to x ∈ (a, b), we obtain that f is also the limit of a sequence of linear combinations of ε α,n , hence L 2 (0, T ) = E(α, T ), which is known to be false since

∞ n=1 1 λ α,n < ∞
(see, e.g., [START_REF] Schwartz | Étude des sommes d'exponentielles[END_REF]).

α,n ) n≥1 in L 2 ((a, b) × (0, T )).
Moreover it is optimal in the following sense: if ( Σm ) m≥1 is another biorthogonal family for the set (ε α,n T,a,b;m . At last, we note that if the sequence of functions ( Σ+ m ) m≥1 is a biorthogonal family for the set (e λα,nt Φ α,n (x)

) n≥1 in L 2 ((a, b)×(0, T )), then for all m ≥ 1, Σ- m -σ- α,T,a,b;m is orthogonal to all εα,n , hence to E(α, T, a, b), hence to σ- α,T,a,b;m since σ- α,T,a,b;m ∈ E(α, T, a, b). Hence Σ- m 2 L 2 ((a,b)×(0,T )) = σ- α,T,a,b;m 2 L 2 ((a,b)×(0,T )) + Σ- m -σ- α,T,a,b;m 2 L 2 (0,T ) ≥ σ- α,T,a,b;m 2 L 2 ((a,b)×(0,T )) . Therefore (5. 7) Σ- m L 2 ((a,b)×(0,T )) ≥ σ - α,T,a,b;m L 2 ((a,b)×(0,T )) = 1 d α,
) n≥1 in L 2 ((a, b) × (0, T )), then T 0 b a Σ+ m (x, T -s)e λα,mT e -λα,ns Φ α,n (x) dx ds = δ m,n , hence ( Σ+ m (x, T -s)e λα,mT
) m is biorthogonal for the set (e -λα,nt Φ α,n (x)) n≥1 in L 2 ((a, b) × (0, T )). This implies that (And of course this lower bound is achieved for the optimal biorthogonal sequence and hence optimal.) Now, comparing with what we noticed in section 5.1, we obtain that (5. 9) ∀m ≥ 1, C N C (α, T ) ≥ e -λα,mT d α,T,a,b;m .

In the x-independent version, we were able to produce precise lower bounds of the right hand-side term, see [START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF][START_REF] Cannarsa | The cost of controlling degenerate parabolic equations by locally distributed controls[END_REF]. But unfortunately, in the present x-dependent case, we were not able to provide the expected property:

∃c u > 0, ∀α ∈ [0, 1), C N C (α, T ) ≥ c u 1 -α e cu T e -1 cu T .
However, in the following we provide independent estimates of the behavior of C N C (α, T ) with respect to α and with respect to T .

5.3.

Optimal lower bound with respect to α.

Here the basic remark is that

d α,T,a,b;1 = dist (ε α,1 , E α,T,a,b;1 ) = dist (ε α,1 , Vect (ε α,n , n ≥ 2) ≤ dist (ε α,1 , εα,2 ) = εα,1 -εα,2 L 2 ((a,b)×(0,T )) .
From the expressions (5. 4) and (4. 4)-(4. 7), we have for t ∈ (0, T ) and x ∈ (a, b):

εα,1 (x, t) = e -κ 2 α j 2 -να ,1 t √ κ α |J -να (j -να,1 )| x 1-α 2 J -να (j -να,1 x κα ), and εα,2 (x, t) = e -κ 2 α j 2 να ,1 t √ κ α |J να (j να,1 )| x 1-α 2 J να (j να,1 x κα ).
These two expressions differ only by the order of the Bessel functions and related zeros: -ν α for εα,1 and ν α for εα,2 . We recall that

(5. 10) ∀ν ∈ [- 1 2 , 1 2 ], ∀y > 0, J ν (y) = ∞ m=0 (-1) m m! Γ(m + ν + 1) y 2 2m+ν .
Then by standard regularity results (smoothness with respect to ν ∈ [- 1 2 , 1 2 ] and y ∈ (0, +∞), implicit value theorem and smoothness of ν → j ν,1 near ν = 0), we have ∃C a,b , sup

x∈[a,b] sup t∈[0,T ] |ε α,1 (x, t) -εα,2 (x, t)| ≤ C a,b (1 -α). Hence d α,T,a,b;1 ≤ C a,b (1 -α),
and

C N C (α, T ) ≥ e -λα,mT C a,b (1 -α) .
This proves the first part of Theorem 2.2. (Note that this was already done in [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF], but in a slightly more elementary approach that was not useful to the exponential behavior, that we prove in the following). (5. 11)

     u t -(|x| α u x ) x = h(x, t)χ ωsym , x ∈ (-1, 1), t > 0, u(-1, t) = 0 = u(1, t), t > 0, u(x, 0) = u 0 (x),
x ∈ (-1, 1).

The associated cost of controllability is

C ctr (α, T ; ω sym ) := sup u0 =1 inf{ h L 2 (ωsym) , u (h) (T ) = 0}.
Of course, given u 0 of L 2 -norm equal to 1, any control h that drives the solution of (1. 1) to 0 in time T gives a control that drives the solution of (5. 11) to 0 in time T , just choosing h = h on (a, b), h = 0 on (-b, -a),

and with this choice we obviously have h

L 2 (ωsym) = h L 2 (ω) .
Hence it is cheaper to control (5. 11) than (1. 1), or, more rigorously:

C ctr (α, T ; ω sym ) ≤ C ctr (α, T ; ω).
Hence, to bound from below C ctr (α, T ; ω), it is sufficient to bound from below C ctr (α, T ; ω sym ).

5.4.b.

A related boundary control problem. Now, consider a control h that drives the solution u h of (5. 11) to 0 in time T , and define

H [u0] -(t) := u h(-a, t), H [u0]
+ (t) := u h(a, t). As recalled in section 4.1.a, we have u ∈ L 2 (0, T ; H 1 α (-1, 1)); but remember that

H 1 α (-1, 1) ⊂ C 0 ([-1, 1]
), the injection being continous; hence u ∈ L 2 (0, T ; C 0 ([-1, 1])), which implies that H

[u0]

-, H

[u0] + ∈ L 2 (0, T ). Denote (5. 12) ∀x ∈ (-a, a), ∀t ∈ (0, T ), v(x, t) := u h(x, t).

Then v satisfies (5. 13)

         v t -(|x| α v x ) x = 0, x ∈ (-a, a), t > 0, v(a, t) = H [u0] + (t), t > 0, v(-a, t) = H [u0] -(t), t > 0, v(x, 0) = u 0 (x),
x ∈ (-a, a),

and

(5. 14) v(T ) = 0.

Hence, roughly speaking, H

[u0] -and H

[u0] + are boundary controls that drive the solution of (5. 13) to 0 in time T . In the following we investigate the associated spectral problem. Consider the associated eigenvalue problem:

(5. 15)

-(|x| α ψ ) = µψ, x ∈ (-a, a), ψ(-a) = 0 = ψ(a).
The solutions of this eigenvalue problem are given in the following: Lemma 5.2. When α ∈ [0, 1) and a ∈ (0, 1), we have exactly two sub-families of eigenvalues and associated eigenfunctions for problem (5. 15), that is:

• the eigenvalues of the form

κ 2 α a 2κα j 2 να,n
associated with the odd function

(5. 16) ψ (o) α,n (x) = x 1-α 2 J να (j να,n ( x a ) κα ) if x ∈ (0, a),
• the eigenvalues of the form κ 2 α a 2κα j 2 -να,n , associated with the even function

(5. 17) ψ (e) α,n (x) = x 1-α 2 J -να (j -να,n ( x a ) κα ) if x ∈ (0, a).
The proof is the same as the proof of Proposition 2.7 of [START_REF] Cannarsa | Null controllability for parabolic operators with interior degeneracy and one-sided control[END_REF], taking into account that here the space domain is (-a, a).

5.4.d. Gap properties of the square roots of the eigenvalues.

It is practical to order the eigenvalues in the increasing order: as we did before, the increasing sequence of eigenvalues is (µ α,n ) n≥1 , where

∀n ≥ 1, µ α,2n-1 = κ 2 α a 2κα j 2 -να,n and µ α,2n = κ 2 α a 2κα j 2 να,n .
We note the following gap estimates:

• concerning consecutive eigenvalues of even order: √ µ α,2n+2 -√ µ α,2n = κ α a κα (j να,n+1 -j να,n ), hence using Komornik-Loreti [START_REF] Komornik | Fourier Series in Control Theory[END_REF] p. 135:

(5. 18) ∀n ≥ 1, √ µ α,2n+2 - √ µ α,2n ≤ κ α a κα π,
• concerning consecutive eigenvalues of odd order: √ µ α,2n+1 -√ µ α,2n-1 = κ α a κα (j -να,n+1 -j -να,n ), hence, in the same way ([24] p. 135):

(5. 19) ∀n ≥ 1, √ µ α,2n+1 - √ µ α,2n-1 ≤ κ α a κα π, • concerning consecutive eigenvalues, we derive that -first √ µ α,2n - √ µ α,2n-1 ≤ √ µ α,2n - √ µ α,2n-2 ≤ κ α a κα π, -and in the same way √ µ α,2n+1 - √ µ α,2n ≤ √ µ α,2n+1 - √ µ α,2n-1 ≤ κ α a κα π.
These gap properties will be important in the following. 5.4.e. Consequences of the moment method for (5. 13).

We use the moment method to obtain useful estimates given by (5. 13) and (5. n are chosen such that ψ α,n L 2 ((-a,a)) = 1 (hence the set {ψ α,n , n ≥ 1} forms an orthonormal basis of L 2 (-a, a)). Next, we denote w α,n (x, t) = ψ α,n (x)e µα,nt , and we see that w α,n is solution of the adjoint problem: w t + (|x| α w x ) x = 0 on (-a, a) × (0, T ), w(-a, t) = 0 = w(a, t).

Then, we multiply the first equation in (5. 13) by w α,n :

0 = T 0 a -a (v t -(|x| α v x ) x w α,n ,
and integrating by parts, we obtain

u 0 , ψ α,n L 2 ((-a,a)) = a α ψ α,n (a) T 0 H [u0] + (t) e µα,nt dt -a α ψ α,n (-a) T 0 H [u0] -(t) e µα,nt dt.
Now, take specifically u 0 := ψ α,1 . Then, choosing p ≥ 1 and n = 2p -1, we have: for all p ≥ 1,

a α ψ α,2p-1 (a) T 0 H [ψα,1] + (t) e µα,2p-1t dt -a α ψ α,2p-1 (-a) T 0 H [ψα,1] - (t) e µα,2p-1t dt = δ 1,2p-1 = δ 1,p .
Since ψ α,2p-1 is an even function, ψ α,2p-1 (-a) = -ψ α,2p-1 (a), and from its formula we have also of course ψ α,2p-1 (a) = 0. Hence, for all p ≥ 1, we have

T 0 H [ψα,1] + (t) e µα,2p-1t dt + T 0 H [ψα,1] - (t) e µα,2p-1t dt = δ 1,p a α ψ α,2p-1 (a) = δ 1,p a α ψ α,1 (a)
.

This implies that

(5. 20) ∀p ≥ 1, T 0 a α ψ α,1 (a)(H [ψα,1] + (t) + H [ψα,1] - (t)) e µα,2p-1t dt = δ 1,p .
In the same way, choosing k ≥ 1 and u 0 := ψ α,2k-1 , we obtain (5. 21)

∀k, p ≥ 1, T 0 a α ψ α,2k-1 (a)(H [ψ α,2k-1 ] + (t) + H [ψ α,2k-1 ] - (t)) e µα,2p-1t dt = δ k,p .
This tells us that the sequence a α ψ α,2k-1 (a)(H

[ψ α,2k-1 ] + + H [ψ α,2k-1 ] - ) k≥1
is biorthogonal to the sequence (e µα,2p-1t ) p≥1 in L 2 (0, T ), and thanks to (5. 19) we are in position to apply Theorem 2.5 of [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], that gives a lower bound for any biorthogonal sequence to a set of exponentials satisfying some gap condition as (5. 19). In our situation, since (5. 19) is true, Theorem 2.5 of [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF] gives us that there exists c u > 0 independent of T , α and k such that

(5. 22) ∀k ≥ 1, a α ψ α,2k-1 (a)(H [ψ α,2k-1 ] + + H [ψ α,2k-1 ] - ) 2
L 2 (0,T )

≥ e -2µ α,2k-1 T e a 2κα 2κ 2 α π 2 T β(T, a, 2k -1), with some β(T, a, 2k -1) explicitly given in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], that behaves in a rational way with respect to T (and then the main behavior as T → 0 is given by the exponential factor e In the following, we conclude using some energy estimates.

5.4.f. Energy estimates. Now we can conclude, relying the locally distributed control (on ω sym ) with the boundary controls acting at -a and a, using energy methods, as we did in [START_REF] Cannarsa | The cost of controlling degenerate parabolic equations by locally distributed controls[END_REF].

As proved in section 5.4.e, H = u h(a, •) + u h(-a, •) has to be exponentially large when T → 0, and this will force h to be also exponentially large. Indeed, first we have ∀y ≥ a, -u(y, t) = Then the null controllability cost for (5. 11) blows up at least exponentially fast when T → 0 + , and as a consequence also for (1. 1), as stated in Theorem 2.2.

Some remarks.

We were not able to use (5. 5) and (5. 9) to prove (2. 4). However, (5. 5) and (5. 9) contain interesting informations, and in the following we give two limit cases. which gives that the null controllability cost blows up at most as 1 T , and not exponentially. Of course this can be proved directly, but the goal of this remark was to obtain this information through (5. 5) and (5. 9). ). That would give (2. 4) in this "limit case".

  +γ * min e -2λmT B(T, γ min , γ * min ), with (2. 8)

3 . 3 3. 1 .

 331 Proof of Theorem 2.The general strategy.

  k ; • the reasoning is symmetric if k < 2m and k is even, hence there at most √ ρ γmin+γ * min such integers k ; • when we consider the case where k > 2m and k is odd, hence k = 2k + 1 with k ≥ m ; in the same way:

(3. 14 ) 2 ∞ 1 )

 1421 P N ,T (z) := e iz T k=N cos(a k z). Then we have the following Lemma 3.4. ([10]) (The regularity and the growth of P N ,T over C: The function P N ,T is entire over C and satisfies (3. 15)

2 L 2 1 + γ min ) 2 e

 2212 (0,T ) ≤ C u 1+ (γ min + γ * min ) 2 γ min ( √ λ -2λmT e Cu √ λm γ min +γ * min B(T, γ min , γ * min ),

  and the operator A : D(A) ⊂ L 2 (-1, 1) → L 2 (-1, 1) defined by D(A) := H 2 α (-1, 1) and ∀u ∈ D(A), Au := (|x| α u x ) x . Then the following results hold: Proposition 4.1. ([7])

  γ min (α)+γ * min (α) e -2λα,mT .

5. 2 .

 2 Classical general considerations. Denote (5. 3) ε α,n : t ∈ (0, T ) → e -λα,nt , and E(α, T ) the smallest closed subspace of L 2 (0, T ) containing all the functions ε α,n with n ≥ 1. Introduce also their x-dependent version: denote (5. 4) εα,n : (x, t) ∈ (a, b) × (0, T ) → e -λα,nt Φ α,n (x), and E(α, T, a, b) the smallest closed subspace of L 2 ((a, b) × (0, T )) containing all the functions εα,n with n ≥ 1.

2 L 2 (

 22 Now, given m ≥ 1, denote E(α, T, a, b; m) the smallest closed subspace of L 2 ((a, b) × (0, T )) containing all the functions εα,n with n ≥ 1, and n = m. Then consider pα,T,a,b;m the orthogonal projection of εα,m on E(α, T, a, b; m), and d α,T,a,b;m the distance between εα,m and E(α, T, a, b; m): we have (5. 5) d 2 α,T,a,b;m = inf p∈E(α,T,a,b;m) εα,m -p (0,T )×(a,b)) λα,ms Φ α,m (x) -pα,T,a,b;m (x, s)) 2 dx ds. Then εα,m -pα,T,a,b;m is orthogonal to E(α, T, a, b; m), which implies that ∀n = m, λα,ms Φ α,m (x) -pα,T,a,b;m (x, s))e -λα,ns Φ α,n (x) dx ds = 0, and T 0 (e -λα,ms Φ α,m (x) -pα,T,a,b;m (x, s))e -λα,ms Φ α,m (x) dx ds = T 0 (e -λα,ms Φ α,m (x) -pα,T,a,b;m (x, s))(e -λα,ms Φ α,m (x) -pα,T,a,b;m (x, s)) ds = d 2 α,T,a,b;m . ,a,b;m (x, s) := e -λα,ms Φ α,m (x) -pα,T,a,b;m (x, s) d 2 α,T,a,b;m : the sequence of functions (σ - α,T,a,b;m ) m≥1 is a biorthogonal family for the set (ε

Hence 1 d

 1 α,T ,a,b;m is a lower bound of every biorthogonal sequence ( Σm ) m≥1 . So a bound from above for d α,T,a,b;m gives a bound from below for every biorthogonal sequence.

2 (

 2 (a,b)×(0,T )) ≥ e -λα,mT d α,T,a,b;m .

5. 4 .

 4 Classical exponential lower bound in short time.5.4.a. Adding a control region.Let us consider the symmetrised control regionω sym := (-b, -a) ∪ (a, b),and the associated control problem:

5. 4

 4 .c. The eigenvalue problem in (-a, a).

  [START_REF] Ervedoza | Sharp observability estimates for heat equations[END_REF]. First we denote∀n ≥ 1, ψ α,2n-1 (x) := C (e)n ψ (e) α,n (x), andψ α,2n (x) := C (o) n ψ (o) α,n(x) in such a way that ψ α,n is an eigenfunction associated to the eigenvalue µ α,n , and where the constants C

a 2κα 2κ 2 α π 2 T 2 .L 2 2κα 4κ 2 α π 2 /2 1 T a 2κα + 1 1

 22222211 ). In particular, in the case k = 1, we have β(T, a, Moreover, since |ψ α,1 (a)| is bounded from below by a positive constant (independent of α ∈ [0, 1) but of course depending on a), we derive from (5(0,T ) ≥ e -µα,1T e a T β(α, T, a) with β(α, T, a) = c u a 2κα T 3a α |ψ α,1 (a)| .

1 yu 1 y 2 ≤ 1 y 1 yx -α dx ≤ 1 a α 1 y x α u 2 x≤ 1 a α y - 1 |x| α u 2 x 1 - 1 u hχ ωsym = T 0 1 - 1 u

 1121112121111 x (x, t) dx, hence ∀y ≥ a, u(y, t) 2 = u x (x, t) dx x α u 2x (x, t) dx (x, t) dx, and in the same way ∀y ≤ -a, u(y, t) 2 (x, t) dx.Then, multiplying the first equation of (5. 11) by u, we haveT 0 (u t -(|x| α u x ) x ) =t) 2 dx dt ≥ a 2α [ u(a, •) 2 L 2 (0,T ) + u(-a, •) 2 L 2 (0,T ) ] -a α ,and the lower bound (5. 23) implies thath 2 L 2 (ω×(0,T )) ≥ a 2α 4 e -2µα,1T e a 2κα 2κ 2 α π 2 T β(α, T, a) 2 -a α .

5. 5 0 e

 50 .a. The limit case a = -1, b = 1. Consider the case of the globally distributed control a = -1, b = 1: thend 2 α,T,-1,1;m = inf p∈E(α,T,-1,1;m) εα,m -p 2 L 2 ((-1,1)×(0,T )) = εα,m 2 L 2 ((-1,1)×(0,T ))since εα,m and E(α, T, -1, 1; m) are orthogonal. Then we immediately haved 2 α,T,-1,1;m = T -λα,mt dt = 1 -e -λα,mT λ α,m ≤ T,

5. 5

 5 .b. The limit case b "close" to a. Let us look to a formula "close" to the one of d 2 α,T,-1,1;m , changing Φ α,n (x) by Φ α,n (a) on the interval (a, b). (Of course this would only have a sense if b = a, which would bring other problems.) Then, if Φ α,n (a) = 0 for all n, we would obtain thatdist 2 L 2 ((a,b)×(0,T )) (ε α,1 Φ α,1 (a), Vect {ε α,n Φ α,n (a), n ≥ 2}) = (b -a)Φ α,1 (a) 2 dist 2 L 2 (0,T ) (ε α,1 , Vect {ε α,n , n ≥ 2}),and then Theorem 2.5 of[START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF], already applied before in section 5.4.e, says that dist L 2 (0,T ) (ε α,1 , Vect {ε α,n , n ≥ 2}) ≤ (1 -α)e -cu T (using Lemmas 5.1 and 5.2 of[START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations[END_REF]

This research was partly supported by the Institut Mathematique de Toulouse and Istituto Nazionale di Alta Matematica. Part of this work was done during the 'Indam intensive period' in Naples in 2019, supported by LIA COPDESC. We thank the organizers for this opportunity. Moreover, the first author acknowledges support by the MIUR Excellence Department Project awarded to the Department of Mathematics,


Antoine Monserand
email: antoine.monserand@univ-paris13.fr

A note on "Zero growth and structural change in a post Keynesian growth model

Keywords: Zero growth, Kaleckian, Stability, Wage-led/Profit-led, Profit rate. JEL Codes: E12, E22, E24, O41, O44

 concerning zero growth and structural change in a post-Keynesian growth model, some of which are shown to be problematic. First, the (im)possibility for a neo-Kaleckian model of growth and distribution to generate a profit-led growth regime is discussed. Next, we review the role played by the "paradox of costs" when introducing the depreciation of capital and how this changes the stability characteristics of the model presented by Rosenbaum. Finally we show that, contrary to what is claimed in the article, the proposed model is not able to show that zero growth is compatible with a positive net rate of profit.

Introduction

In an article published in 2015 and titled "Zero growth and structural change in a post Keynesian growth model", Rosenbaum investigates in a Kaleckian framework the possibility and conditions for a state of zero growth to be stable. We believe that this question is of high importance, given the negative impacts of economic growth on the environment, and therefore on humans and other species. The field of "post-Keynesian ecological macroeconomics" has recently gained momentum [START_REF] Fontana | Towards Post-Keynesian Ecological Macroeconomics[END_REF] and the article by [START_REF] Rosenbaum | Zero Growth and Structural Change in a Post Keynesian Growth Model[END_REF] contributes to its advancement. However, several results found in it seem problematic and are analysed in this note. First, the (im)possibility for a neo-Kaleckian model of growth and distribution to generate a profit-led growth regime is discussed. Next, we review the role played by the "paradox of costs" when introducing the depreciation of capital and how this changes the stability characteristics of the model presented by Rosenbaum. Finally we show that, contrary to what is claimed in the article, the proposed model is not able to show that zero growth is compatible with a positive net rate of profit.

Profit-led growth in a neo-Kaleckian model

Rosenbaum builds a neo-Kaleckian model, with the following specifications for the saving and investment functions (the numbers in square brackets refer to the numeration in the article of Rosenbaum):

𝜎𝜎 = 𝑠𝑠 ⋅ 𝑟𝑟 𝑛𝑛 [16];(1) 𝑔𝑔 = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝜏𝜏𝑟𝑟 𝑛𝑛 + 𝜑𝜑𝜑𝜑 [17];(2)
where 𝜎𝜎 and 𝑔𝑔 are the ratios of net saving and of net investment to the capital stock, 𝛽𝛽 is the rate of capacity utilisation, 𝑟𝑟 𝑛𝑛 is the rate of profit net of depreciation and 𝜑𝜑 is the rate of technical progress. The parameter 𝑠𝑠 represents the propensity to save out of profits ; the propensity to save out of wages is assumed to be equal to zero. With the rate of capital depreciation being proportional to technical progress 𝜑𝜑, the net rate of profit is given by:

𝑟𝑟 𝑛𝑛 = 𝜋𝜋𝛽𝛽 𝜈𝜈 -𝑑𝑑 1 𝜑𝜑 [13];(3)
where 𝜋𝜋 stands for the profit share and 𝜈𝜈 for the capital to capacity ratio.

As usual with this type of model, these three equations lead to the equilibrium rate of accumulation 𝑔𝑔 ⋆ :

𝑔𝑔 ⋆ = 𝑠𝑠 � 𝜋𝜋 𝜈𝜈 � 𝛼𝛼 + (𝑠𝑠 -𝜏𝜏)𝑑𝑑 1 𝜑𝜑 + 𝜑𝜑𝜑𝜑 (𝑠𝑠 -𝜏𝜏)(π/ν) -𝛽𝛽 -𝑠𝑠𝑑𝑑 1 𝜑𝜑 [21];(4)
From this, Rosenbaum argues that the effect of a change in the profit share 𝜋𝜋 on 𝑔𝑔 ⋆ "depends on the specific parameter configuration", and therefore growth can be either profit-driven or wage-driven. This is at odds with the usual results for neo-Kaleckian models of growth and distribution in a closed economy with no saving out of wages -which is the case here: they can only show wage-led growth (Hein 2014, p. 271). However, since Rosenbaum's model is more complex as it features productivity growth and capital depreciation, it might be the case that a profit-led growth regime is possible. But instead of leaving the question open and examining the properties of this regime, its plausibility should first be examined. In the rest of this section, we do this check and show that the profit-led growth regime is implausible in Rosenbaum's model.

Computing the partial derivative of 𝑔𝑔 ⋆ with respect to 𝜋𝜋, one obtains:

𝜕𝜕𝑔𝑔 ⋆ 𝜕𝜕𝜋𝜋 = -𝑠𝑠𝛽𝛽 𝛼𝛼 + (𝑠𝑠 -𝜏𝜏)𝑑𝑑 1 𝜑𝜑 + 𝜑𝜑𝜑𝜑 𝜈𝜈[(𝑠𝑠 -𝜏𝜏)(𝜋𝜋/𝜈𝜈) -𝛽𝛽] 2 (5)
The sign of this expression depends on the signs of 𝑠𝑠𝛽𝛽 and on that of the numerator of the fraction. Firstly, Rosenbaum implicitly assumes the propensity to save out of profits to be positive, since he looks at cases where the goods market equilibrium is stable and this can only happen if the propensity 𝑠𝑠 is strictly positive. The stability of the goods market is derived from the assumption that saving is more reactive than investment to changes in the rate of capacity utilisation, which is known as the "Keynesian stability condition". In Rosenbaum's model, this condition is given by equation ( 6) and requires that 𝑠𝑠 be strictly positive:

(𝑠𝑠 -𝜏𝜏)(𝜋𝜋/𝜈𝜈) -𝛽𝛽 > 0 [20];(6)
Secondly, 𝛽𝛽 must be positive for the model to make economic sense, since it represents the sensitivity of investment to the rate of capacity utilisation 1 . Rosenbaum supports this view himself, as can be seen from his discussion on page 633 where he discards his "Case 1" precisely because it requires 𝛽𝛽 to be negative. Finally, could the numerator of the fraction in equation ( 5) be negative? Looking at Rosenbaum's equation [18], the expression for the equilibrium rate of capacity utilisation 𝛽𝛽 ⋆ :

𝛽𝛽 ⋆ = 𝛼𝛼 + (𝑠𝑠 -𝜏𝜏)𝑑𝑑 1 𝜑𝜑 + 𝜑𝜑𝜑𝜑 (𝑠𝑠 -𝜏𝜏)(π/ν) -𝛽𝛽 [18];(7)
we see that this cannot be the case. Otherwise, 𝛽𝛽 ⋆ would be negative since the numerator is the same as the one from equation ( 5) and the denominator is assumed to be positive for the Keynesian stability condition to hold.

Thus, the model used by Rosenbaum does not allow for any profit-led growth regime, and thereby the whole discussion of pages 636-40 on the profit-driven case and its (in)stability is in fact irrelevant.

Depreciation and the paradox of costs

The second substantial problem we find in Rosenbaum's article is in the effect of the rate of depreciation of capital on the rate of accumulation. According to him, a higher (lower) rate of depreciation entails a lower (higher) rate of accumulation, seemingly because it would decrease (increase) net investment, for a given amount or rate of gross investment. This is what we understand from footnote 4 of his article, where he discusses a departure from a zero growth state:

Of course, if the actual depreciation factor is set at a lower value, growth would be positive since investment now exceeds depreciation whereas for higher depreciation factors, growth would become negative. [...] In Figure 3 and Figure 4, different depreciation factors would be captured by shifting the 𝑔𝑔 and 𝜎𝜎 schedules upward or downward considering that the depreciation factor 𝑑𝑑 1 enters as a constant in Equations ( 16) and ( 17). (Rosenbaum 2015, p. 633) Equations [16] and [17] are the saving and investment functions of the model, both of which indeed contain the depreciation factor as a constant, coming from the rate of profit net of depreciation. But Rosenbaum is mistaken in assuming that both the the saving (𝜎𝜎) and investment (𝑔𝑔) schedules shift upward or downward together by the same distance when the depreciation factor is included (or changes). In his Figure 3, Rosenbaum presumes that both schedules shift down by a height 𝑑𝑑 due to depreciation, therefore allowing a zero growth equilibrium to be reached while maintaining a constant rate of capacity utilisation.

In fact, the coefficients in front of the depreciation term 𝑑𝑑 1 𝜑𝜑 in Rosenbaum's equations [16] and [17] are not the same: it is -𝑠𝑠 in the saving equation (𝑠𝑠 being the propensity to save out of profits), and -𝜏𝜏 in the investment equation (𝜏𝜏 being the sensitivity of investment to the net rate of profit). The assumption made that the Keynesian stability condition holds necessarily implies that 𝑠𝑠 is bigger than 𝜏𝜏. Hence, if the depreciation factor 𝑑𝑑 1 increases (decreases), the saving curve 𝜎𝜎 will shift downward (upward) by a larger amount than the investment curve 𝑔𝑔. This is not anecdotal, since instead of reducing the equilibrium rate of accumulation while keeping the rate of capacity utilisation constant, as is shown by Rosenbaum, an increase in the depreciation factor would lead to an increase in the equilibrium rate of accumulation, as well as in the rate of capacity utilisation.

INSERT FIGURE 1 HERE OR NEARBY

CAPTION FOR FIGURE 1: Change in the equilibrium rates of accumulation and of capacity utilisation when the rate of capital depreciation is introduced.

Figure 1 illustrates this phenomenon: the solid lines 𝑔𝑔 𝑠𝑠 and 𝑔𝑔 𝑖𝑖 represent the situation of departure (without depreciation costs); the dotted lines 𝑔𝑔 𝑠𝑠′ and 𝑔𝑔 𝑖𝑖′ show the situation after taking depreciation into account, as is proposed by Rosenbaum (the value of the depreciation factor is set in such a way that the equilibrium rate of growth is zero), and the dashed line 𝑔𝑔 𝑖𝑖″ and dotted line 𝑔𝑔 𝑠𝑠″ illustrate the situation when depreciation is properly taken into account (with the same value for the depreciation factor as in Rosenbaum's proposal).

What we observe in the correct representation is the well-known "paradox of costs" [START_REF] Rowthorn | Demand, Real Wages and Economic Growth[END_REF], applied here to depreciation costs 2 , that plays the decisive role. In order to meet a given level of aggregate demand, firms need to maintain a certain level of productive capital. If depreciation accelerates, gross investment will have to increase in order to compensate. This higher gross investment generates additional economic activity and therefore a higher rate of capacity utilisation, triggering in turn an increase in investment and in the rate of accumulation.

It should be noted that the mechanism described above (that has to do with the shift of the saving curve) more than compensates the disincentive to invest caused by the initial reduction in the net rate of profit (the shift of the investment curve). The overall effect can be checked mathematically by taking the first derivative of the equilibrium rate of accumulation 𝑔𝑔 ⋆ with respect to the depreciation factor 𝑑𝑑 1 , using equation ( 4):

𝜕𝜕𝑔𝑔 ⋆ 𝜕𝜕𝑑𝑑 1 = 𝑠𝑠𝛽𝛽𝜑𝜑 (𝑠𝑠 -𝜏𝜏)(π/ν) -𝛽𝛽 (8)
This expression cannot but be positive, considering our previous remarks on the signs of 𝛽𝛽 and of the denominator, plus the fact that the rate of technical progress 𝜑𝜑 has to be positive for all the discussions concerning this article. Indeed, from Rosenbaum's saving equation [16] we see that the equilibrium rate of capacity utilisation 𝛽𝛽 ⋆ , in the case of zero growth, is equal to 𝜈𝜈𝑑𝑑 1 𝜑𝜑 𝜋𝜋 ⁄ . Thus, 𝜑𝜑 > 0 is required in order to ensure 𝛽𝛽 ⋆ > 0.

Implications for the discussion on stability

Several implications that Rosenbaum claims from his model have to do with the stability analysis of a zero growth economy in various configurations. He distinguishes the cases of "laissez-faire" and "forced" zero growth on one hand, and of profit-driven and wage-driven growth on the other hand, as summarised in his Table 1. As we have already argued, the profit-driven cases are irrelevant for this model. But what of the stability analysis conducted for a wage-driven economy?

According to Rosenbaum, a wage-driven economy is generally unstable. For instance it is destabilised by an increase in the markup, because of a spiral of falling wages and employment in the absence of automatic stabilisers. But by forcing zero growth through a rapid monitoring of the depreciation factor, the unstable economy is said to be stabilised. The idea is that following an increase in the markup and therefore a fall in the rate of growth, public authorities would change tax rules and/or technical regulations and norms so that the depreciation factor 𝑑𝑑 1 would decrease, in an attempt to increase the rate of growth. Rosenbaum describes the stabilizing effect in the following way: because depreciation is lower, renewal of capital is slower. As a result, technical progress is slowed, meaning less labour is "saved". This has a positive effect on employment and wages, which in turn tends to increase growth.

One could doubt the strength of this stabilising effect, since the effect of a slight acceleration or deceleration of technical progress on overall employment, and above all the subsequent effect on wages, seems to be of secondary order in our view. But more importantly, the mechanism described here does not work the same way once we take into account the paradox of costs mentioned above. Indeed, after an increase in the markup and a fall in the rate of growth, public authorities should now try to raise the rate of depreciation, since this would bring about higher growth. By doing so, however, the mechanism is reversed: faster renewal of capital would speed up labor saving processes, reducing employment. Wages would fall along with the rate of growth. In the terms of the article's narrative, this would be an unstable economy, and we should then conclude that a wage-driven economy cannot be stable at zero growth, no matter how "free" or "forced" this zero growth is. Future work should put forward other stabilising mechanisms that would allow for a wage-led economy to be stable at zero growth.

Zero growth and the net rate of profit

Our last concern has to do with the discussion on the net rate of profit in the case of zero growth presented in the section "Further considerations" of Rosenbaum's article, on page 643. Starting from the definition of the rate of profit net of depreciation, inserting the condition for zero growth regarding the depreciation factor and using the expression for the equilibrium rate of capacity utilisation, Rosenbaum arrives at a surprising condition which if verified allows for the net rate of profit to be positive while the rate of growth is zero. The condition is the following:

𝛽𝛽 𝜈𝜈 𝜋𝜋 (1 -𝛽𝛽 2 ) (1 -𝛽𝛽) < 𝑠𝑠 -𝜏𝜏 (9)
We have not ascertained whether this result is due to a typo in Rosenbaum's equation [33] where, on its right-hand side, a coefficient 𝛽𝛽 is missing in front of 𝛽𝛽 in the numerator, or is due to some erroneous calculation, but we can assert that the result is incorrect. Indeed, the correct expression for the net rate of profit is:

𝑟𝑟 𝑛𝑛 = 𝛽𝛽𝛽𝛽 + 𝛼𝛼 + 𝜑𝜑𝜑𝜑 𝛽𝛽ν/π (10)
Replacing 𝛽𝛽 with its value at the zero growth equilibrium and then using Rosenbaum's zero growth condition [23] on 𝑑𝑑 1 , we get:

𝑟𝑟 𝑛𝑛 = 𝛽𝛽𝑑𝑑 1 𝜑𝜑𝜈𝜈/𝜋𝜋 + 𝛼𝛼 + 𝜑𝜑𝜑𝜑 𝛽𝛽𝜈𝜈/𝜋𝜋 = -𝛼𝛼 -𝜑𝜑𝜑𝜑 + 𝛼𝛼 + 𝜑𝜑𝜑𝜑 𝛽𝛽𝜈𝜈/𝜋𝜋 = 0 (11)
This result is unsurprising and can be obtained even more easily, by directly using the saving equation ( 1) taken with 𝜎𝜎 = 0, since at equilibrium 𝜎𝜎 = 𝑔𝑔, yielding:

0 = 𝑠𝑠 ⋅ 𝑟𝑟 𝑛𝑛 (12)
Therefore 𝑟𝑟 𝑛𝑛 = 0 since 𝑠𝑠 ≠ 0 here 3 . Thus we show that the model presented by Rosenbaum does not prove that a positive rate of net profit is compatible with zero growth. This compatibility, however, is a standard feature of stock-flow consistent models and has been put forward more explicitly in recent articles related to the "monetary growth imperative" controversy (Cahen-Fourot and Lavoie 2016; [START_REF] Jackson | Does Credit Create a Growth Imperative? A Quasi-Stationary Economy with Interest-Bearing Debt[END_REF].

Conclusion

In this note, we have formulated several critiques regarding Rosenbaum's article. First, we showed that the model presented cannot produce the profit-driven growth regime the author studies in parallel to the wage-driven one. Then we demonstrated that, because of the phenomenon of the "paradox of costs", the only remaining stable case (namely the "forced zero growth" wage-driven regime) is in fact unstable.

Finally we pointed out that, contrary to the author's claim, the model proposed is not able to show that zero growth is compatible with a positive net rate of profit.

As a concluding remark, we shall stress that understanding the conditions under which modern economies can be stable and provide employment and well-being for all without growth remains an important challenge for the emerging field of (post-Keynesian) ecological macroeconomics.

FOOTNOTES:

1: As a limit case, one could consider the specification 𝛽𝛽 = 0. This leads equation ( 5) to be equal to zero and thus also rules out the possibility of profit-led growth.

2: [START_REF] Rowthorn | Demand, Real Wages and Economic Growth[END_REF] presents the paradox of costs and shows that the phenomenon holds for any "real cost of production" (wages, fixed capital requirements, taxes, and capital depreciation).

3: In the particular case where 𝑠𝑠 = 0, the net rate of profit could be positive.

However, as we have argued in the first section, this case is implicitly dismissed by Rosenbaum. Indeed he assumes that the Keynesian stability condition is verified, and in his model this is only possible when 𝑠𝑠 > 0.