Damien Allonsius
email: damien.allonsius@univ-amu.fr

Franck Boyer
email: franck.boyer@math.univ-toulouse.fr

M Morancey
email: morgan.morancey@univ-amu.fr

ANALYSIS OF THE NULL-CONTROLLABILITY OF DEGENERATE PARABOLIC SYSTEMS OF GRUSHIN TYPE VIA THE MOMENTS METHOD

Keywords: Control theory, degenerate parabolic partial differential equations, minimal null control time, moments method, biorthogonal families, spectral analysis of harmonic oscillator AMS subject classifications. 93B05, 93C20, 30E05, 35P20

come

Analysis of the null-controllability of degenerate parabolic systems of Grushin type via the moments method

Introduction.

1.1. The problem under study. Let T > 0, Ω := (-1, 1) × (0, 1) and ω be an open subset of Ω. Let D be the degenerate elliptic operator D := -∂ xx -x 2 ∂ yy . We are interested in the distributed null-controllability properties of the associated parabolic equation (Grushin's equation) with Dirichlet boundary condition, that reads      ∂ t f (t, x, y) + Df (t, x, y) = 1 ω u(t, x, y), (t, x, y) ∈ (0, T ) × Ω f (t, x, y) = 0, (t, x, y) ∈ (0, T ) × ∂Ω f (0, x, y) = f 0 (x, y) ∈ L 2 (Ω), (x, y) ∈ Ω.

(1.1)

More precisely, given T > 0 we will say that (1.1) is null controllable at time T if for any f 0 ∈ L 2 (Ω) there exists a control u ∈ L 2 ((0, T ) × Ω) such that ∀(x, y) ∈ Ω, f (T, x, y) = 0. It turns out that when ω = (a, b) × (0, 1) where 0 < a < b ≤ 1, see Figure 1.1, there exists a positive minimal null control time, that is, a time T * > 0 such that • when T ≤ T * , equation (1.1) is not null controllable at time T ,

• when T > T * , equation (1.1) is null controllable at time T . More precisely, it was proved in [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF] that T * ≥ a 2 2 and, very recently, the following sharp result was obtained in [START_REF] Beauchard | Minimal time issues for the observability of Grushintype equations[END_REF].

Theorem 1.1. Let 0 < a < b ≤ 1. The minimal null control time of (1.1) with ω = (a, b) × (0, 1) is exactly given by The first motivation of our work is to give another proof of this result using moments method. We will take the opportunity of this work to develop new estimates for biorthogonal sequences to real exponentials (see Theorem 1.3) that may have a wider range of applications. Our approach will also enable us to treat coupled systems of such Grushin equations (see Theorem 1.4).

T * = a 2 2 .
Let Γ 1 = {1} × [0, 1]. With classical arguments (see [18, Appendix A] for details in this setting), based on extensions of the spatial domain and cut-off functions, it directly comes from Theorem 1.1 that the minimal null control time for the boundary control problem

        
∂ t f (t, x, y) + Df (t, x, y) = 0, (t, x, y) ∈ (0, T ) × Ω f (t, x, y) = 0, (t, x, y) ∈ (0, T ) × (∂Ω\Γ 1 ) f (t, x, y) = u(t, y), (t, x, y) ∈ (0, T ) × Γ 1 f (0, x, y) = f 0 (x, y) ∈ L 2 (Ω), (x, y) ∈ Ω, (

is given by T * = 1 2 . We will prove directly this result since it extends at no cost to coupled systems of such Grushin equations (see Theorem 1.5) with a single boundary control. To the best of our knowledge, this result is out of reach with other techniques.

State of the art.

The counter-example proving that (1.1) is not null controllable in time T ≤ a 2 2 has been exhibited in the pioneering work [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF]Section 3.3]. We will thus focus on the positive null controllability result in time T > a 2 2 . In [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF] the value of the minimal null control time T * has been proved to be exactly a 2 2 when the control domain is symmetric with respect to the degeneracy i.e. ω = (-b, -a) ∪ (a, b) × (0, 1). In [START_REF] Beauchard | Minimal time issues for the observability of Grushintype equations[END_REF], it has been recently proved that T * := a 2 2 also when ω = (a, b) × (0, 1), that is Theorem 1.1. We mention that the results obtained in [START_REF] Beauchard | Minimal time issues for the observability of Grushintype equations[END_REF] cover more general situations than the one considered in Theorem 1.1: non symmetric space domains, higher dimension (in the y variable), Dirichlet or Neumann controls and more general degeneracies.

Let us mention that, as detailed in [START_REF] Khodja | Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems[END_REF]Section 4.1.3], the reason for the appearance of such a minimal time, is the insufficient observation of some eigenvectors of D that localizes near the singular set {x = 0} too quickly with respect to the desired control time.

The appearance of a minimal null control time in this context is very much related to the geometry of ω. Indeed, it is proved in [START_REF] Koenig | Non-null-controllability of the Grushin operator in 2D[END_REF] that if Ω\ω contains an horizontal strip (-1, 1)×(a, b), then equation (1.1) is not null controllable whatever the time T is.

The proof is done by contradiction using for instance Runge's approximation theorem with quite subtle estimates coming from complex analysis. An (almost) complete study of the value of the minimal null control time with more complex geometries for ω is done in [START_REF] Duprez | Control of the Grushin equation: non-rectangular control region and minimal time[END_REF].

The common point between the results obtained in [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF][START_REF] Beauchard | Minimal time issues for the observability of Grushintype equations[END_REF][START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF] and the analysis proposed in this article is to take advantage of the cartesian geometry of Ω and ω to reduce the study to the one of a family of one dimensional problems. To be more precise, let us introduce the operators

A n := -∂ xx + (nπx) 2 , D(A n ) := H 2 (-1, 1) ∩ H 1 0 (-1, 1),
and let φ n be the n th normalized eigenfunction of Dirichlet Laplace operator on (0, 1):

φ n (y) := √ 2 sin(nπy), y ∈ (0, 1).

Then, as proved in [8, Section 2.3], for any u ∈ L 2 ((0, T ) × ω), expanding u(t, x, y) = n∈N * u n (t, x)φ n (y), (1.3) we get that the solution f of (1.1) can be written as

f (t, x, y) = n∈N * f n (t, x)φ n (y),
where f n is the solution of the following one dimensional non-degenerate parabolic equation

     ∂ t f n (t, x) + A n f n (t, x) = u n (t,
x)1 (a,b) (x), (t, x) ∈ (0, T ) × (-1, 1), f n (t, ±1) = 0, t ∈ (0, T ),

f n (0, x) = f 0 n (x),
x ∈ (-1, 1).

(1.4)

The null controllability of (1.4) is well known (see for instance [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]) and holds for any T > 0. Thus, proving null controllability of (1.1) amounts to estimate a control u n realizing null controllability of (1.4) with respect to n. Indeed, if for a given T > 0, the cost of null controllability of (1.4) is bounded with respect to n i.e. if there exists C > 0 such that for any n and any f 0 n ∈ L 2 (-1, 1) there exists a control u n ∈ L 2 ((0, T ) × (a, b)) satisfying

u n L 2 ((0,T )×(a,b)) ≤ C f 0 n L 2 (-1,1) ,
we will get that (1.3) actually defines a control u ∈ L 2 ((0, T ) × ω) for our initial problem and thus gives null controllability of (1.1) in time T . In [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF][START_REF] Beauchard | Minimal time issues for the observability of Grushintype equations[END_REF], the study of the cost of controllability of (1.4), or equivalently the cost of observability of the adjoint problem, is done using Carleman estimates with adapted weights. In [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF], this analysis relies on the transmutation that reduces the problem to estimating the cost of observability of the wave equation associated with A n . This study is then done, as the equation is one dimensional, using sideways energy estimates (i.e. energy estimates where the role of the time and space variables are interchanged).

In this article we propose a new strategy, based on the the moments method, allowing to recover Theorem 1.1 that we describe in the next section.

Our method.

As already mentioned, we reduced our problem to the one of proving that, when T > a 2 2 , the cost of null-controllability of (1.4) is bounded with respect to n. We will achieve this objective by using the moments method, that is why we first recall the construction of a null control of (1.4) with this strategy.

For any n ≥ 1, we denote by Λ n := (λ k,n ) k≥1 the increasing sequence of eigenvalues of the operator A n . We also denote by (g k,n ) k≥1 the associated Hilbert basis of L 2 (-1, 1) of eigenfunctions and by e -•An the semigroup generated by -A n .

Let u n ∈ L 2 ((0, T ) × (a, b)) and f n be a solution of (1.4). Testing the equation against the solution t → e -(T -t)An g k,n = e -(T -t)λ k,n g k,n of the backward problem, we obtain the following identity

T 0 u n (t, •), e -λ k,n (T -t) g k,n L 2 (a,b) dt = f n (T, •), g k,n L 2 (-1,1) -f 0 n , e -λ k,n T g k,n L 2 (-1,1) .
As (g k,n ) k≥1 is an Hilbert basis of L 2 (-1, 1), it comes that f n (T, •) = 0 if and only if the control u n satisifes the following moments problem: for any k ∈ N * ,

T 0 u n (t, •), e -λ k,n (T -t) g k,n L 2 (a,b) dt = -f 0 n , e -λ k,n T g k,n L 2 (-1,1) . (1.5)
Following for instance [START_REF] Pöschel | Inverse spectral theory[END_REF]Theorem 4], it comes that for any n ∈ N * , k∈N * 1 λ k,n < +∞. Thus, using for instance [28, §7 and §12], we have that, for any n ≥ 1, there exists a biorthogonal family to the exponential functions associated to Λ n i.e. a family (q j,n ) j∈N * ⊂ L 2 (0, T ; R) such that T 0 e -λ k,n t q j,n (t)dt = δ k,j , ∀k, j ∈ N * .

Thus, following the strategy proposed in [START_REF] Lagnese | Control of wave processes with distributed controls supported on a subregion[END_REF], the control

u n (t, x) = - k∈N * e -λ k,n T f 0 n , g k,n L 2 (-1,1) g k,n (x) g k,n 2 L 2 (a,b) q k,n (T -t), (1.6)
formally solves (1.5). It remains to prove that the above series actually converges and to get a bound on u n which is uniform with respect to n. Thus, the two main issues will be to give precise estimates (with respect to k and n) of the following type

• a lower bound on g k,n L 2 (a,b) ; • an upper bound on q k,n L 2 (0,T ) .
The first estimate is given in the following proposition. Proposition 1.1. Let a > 0. For any η > 0 sufficiently small (depending only on a), there exists C η > 0 such that for any k, n ∈ N * ,

1 a g 2 k,n (x)dx ≥ C η e -nπ(a 2 +η) n 2 π 2 . (1.7)
This estimate is a straightforward application of [8, Proposition 4.5] (see for instance [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF]Section 4.3]) and will not be detailed. The only difference is that the results in [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF] give a lower bound on the L 2 (-1, -a) ∪ (a, 1) norm. As the considered eigenfunctions are even or odd (see Lemma 2.1) this directly implies Proposition 1.1. Note that (1.7) is an estimate of g k,n on L 2 (a, 1) and not L 2 (a, b). As mentioned in section 4.1, one can carry the analysis on the control domain (a, 1) × (0, 1) then use a cut-off argument to prove the desired controllability property on (a, b) × (0, 1).

The second estimate will require a careful spectral analysis of the operator A n . In this direction, let us recall the classical result obtained in [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF].

Let ρ > 0 and N : R + → N. For an increasing sequence of positive numbers Θ = (θ k ) k≥1 such that k 1 θ k < +∞ we say that Θ belongs to the class L(ρ, N ) if

θ k+1 -θ k > ρ, ∀k ∈ N, (1.8)
and

k≥N (ε) 1 θ k ≤ ε, ∀ε > 0. (1.9)
With such class of sequences H.O. Fattorini and D.L. Russell proved the following result (see [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF]Theorem 1.5]). Theorem 1.1. Let ρ > 0 and N : R + → N. For any T > 0, for any ε > 0, there exists a constant C ε,T,ρ,N > 0 depending only on ε, T, ρ and N such that, for any sequence Θ ∈ L(ρ, N ), there exists a biorthogonal family (q σ ) σ∈Θ to the family of exponentials (t → e -θt ) θ∈Θ in L 2 (0, T ; R) satisfying

q σ L 2 (0,T ;R) ≤ C ε,T,ρ,N e εσ , ∀σ ∈ Θ.
In order to apply this result to estimate the control (1.6) achieving null controllability of (1.4), uniformly with respect to n, one should at least prove the following uniform (with respect to n) gap property

inf n∈N * inf k∈N * |λ k+1,n -λ k,n | > 0.
(1.10)

Even though numerical results seem to confirm that this property holds true (see Figure 1.2), we were not able to prove it for the whole spectrum. Therefore, in order to conclude to the null controllability of (1.1) in optimal time, we will generalize the estimate of biorthogonal families given in Theorem 1.1 under weaker assumptions. This result is stated and discussed in Section 1.4.2 and may have its own interest in other situations. The different spectral results obtained in the analysis of such property are presented in the next subsection.

Notation.

• In all this article, C denotes a positive number that may vary from line to line. If necessary, we add information about dependency on parameters p 1 , p 2 , . . . , p k in the following way: C p1,p2,...,p k . • For any α, β ∈ R, we will use the following notation:

α, β = N ∩ [α, β].
• We shall use the Kronecker symbol δ x,y which is 1 if x = y and 0 otherwise.

• The real part (resp. imaginary part) of any complex number z ∈ C will be denoted by z (resp. z).

Results.

As already said in subsection 1.3, one of the main achievements concern the spectral analysis of operator A n . The results are presented in Section 1.4.1 and proved in Section 2.

We also extend the results of [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] to obtain uniform estimates on biorthogonal families to real exponentials in the absence of a uniform gap property. This analysis, which has its own interest, is presented in Section 1.4.2 and the detailed proofs are given in Section 3.

Gathering these results, we finally obtain the null controllability of (1.1) for T > a 2

2 using the moments method. This result is extended to cascade systems of Grushin equations. The Theorem is stated in Section 1.4.3 and proved in Section 4.

Results on the spectrum of A n .

To analyze the uniform gap property we have performed numerical simulations. In Figure 1.2 we represent, for different values of n, the gap λ k+1,n -λ k,n divided by nπ, as a function of k. It clearly appears that the uniform gap property (1.10) seems to hold true, and in fact even more since we observe that |λ k+1,n -λ k,n | ≥ 2nπ for any k, n. More precisely, this (normalized) gap appears to be roughly constant equal to 2 for low frequencies whereas it presents a linear growth in k for high frequencies. Moreover, we observe that the position of the transition point between those two regimes seems to grow linearly with respect to the parameter n. To carefully analyze this gap property we will use two different strategies for low frequencies and high frequencies. Let us precisely define those two regimes: for any τ ∈ (0, 1) and n ∈ N * , we define a lower and an upper range of indices by

L τ,n := k ∈ N * : k ≤ τ π 2 n and U n := k ∈ N * : k ≥ nπ 2 + 1 . (1.11)
The parameter τ is intended to be chosen close to 1 (see (4.5) in the controllability proof). Theorem 1.2. There exists γ > 0 such that for any τ ∈ (0, 1) there exists n 0 := n 0 (τ ) ∈ N * such that the following gap property holds

λ k+1,n -λ k,n > γ, ∀k ∈ L τ,n ∪ U n , ∀n ≥ n 0 .
Since for any n ≤ n 0 (τ ) each sequence of eigenvalues (λ k,n ) k∈N * satisfies a gap property depending on n, the next result follows directly.

Corollary 1.2. For any τ ∈ (0, 1), there exists γ τ > 0 such that, the following gap property holds

λ k+1,n -λ k,n > γ τ , ∀k ∈ L τ,n ∪ U n , ∀n ≥ 1.
(1.12)

As highlighted by Figure 1.2, the whole spectrum Λ n seems to enjoy a uniform gap property. However, we were not able to prove it for the indices in the intermediate set

M τ,n := N * \ (L τ,n ∪ U n ) ,
and obtained the much weaker estimate below, that will also play an important role in the analysis.

Proposition 1.2. Let τ ∈ 3 4 , 1 . For any n ∈ N * , we have

λ k,n ≥ n 2 π 2 4 , ∀k ≥ τ n π 2 , (1.13)
and

|λ k,n -λ j,n | ≥ λ k,n 2 exp - 8 3 λ k,n , ∀k ≥ τ n π 2 , ∀j = k. (1.14)
The last ingredient needed is the fact that, the number of eigenvalues in this intermediate region, even if it increases with n, can be chosen to be negligible in some sense since τ can be arbitrarily close to 1. More precisely, we have

#M τ,n ≤ (1 -τ ) nπ 2 + 1, ∀τ ∈ (0, 1), ∀n ≥ 1. (1.15)
The key point to prove Theorem 1.1 (see Section 4) will be that there are not too much eigenvalues in the intermediate range: the factor (1 -τ ) will be crucial since we will be able to choose it as small as necessary.

1.4.2. Existence and estimates of a biorthogonal family to the exponentials.

As the uniform gap property stated in Theorem 1.2 is not established for the whole spectrum, one cannot apply Theorem 1.1 to obtain uniform estimates of the biorthogonal family to the associated exponentials. To cope with this difficulty, we extend the uniform estimates obtained in [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] under weaker assumptions.

Let us first introduce a few definitions. For any countable family Λ ⊂ (0, +∞), we denote by N its counting function defined by N (r) := # {λ ∈ Λ ; λ ≤ r} , ∀r ∈ R.

(1.16)

Moreover for any λ ∈ Λ we define

ρ λ := dist (λ, {0} ∪ Λ\{λ}) , (1.17)
and for any parameter γ > 0, we set

c γ λ := # Λ\{λ} ∩ (λ -γ, λ + γ) . (1.18)
To simplify a little the notation, we do not explicitly mention that those three definitions depend on the whole family Λ.

Observe that

c γ λ = 0, ∀λ ∈ Λ ⇐⇒ inf λ,µ∈Λ λ =µ |λ -µ| ≥ γ, (1.19)
so that c γ λ can be interpreted as a local measure of the possible defect of a uniform gap with parameter γ for the family Λ.

Theorem 1.3. Let T > 0, α ∈ (0, 1) and N > 0. Let Λ ⊂ (0, +∞) be a countable family such that its counting function N satisfies

N (r) ≤ N r α , ∀r > 0. (1.20)
Then, there exists a biorthogonal family (q λ ) λ∈Λ to the functions

t → e -λt ; λ ∈ Λ , in L 2 (0, T ; R) i.e.
T 0 e -λt q µ (t)dt = δ λ,µ , ∀λ, µ ∈ Λ, such that for any λ ∈ Λ, the following estimate holds, for any γ > 0,

q λ L 2 (0,T ) ≤ C T,α, N λ + γ ρ λ c γ λ exp C α, N (1 + log(λ/γ))λ α , (1.21)
where C α, N and C T,α, N only depend on the parameters α, N (and T ).

In Section 3, this result is extended to biorthogonal families to t → (-t) p e -λt ; p ∈ 0, d , λ ∈ Λ .

For more details we refer to Theorem 3.2.

The existence of a biorthogonal family as stated in Theorem 1.3 is classical (see for instance [START_REF] Schwartz | Étude des sommes d'exponentielles réelles[END_REF]). The main novelty is thus the estimate (1.21) which do not require any gap-type assumption and whose dependency with respect to the family Λ is carefully given.

Let Λ be a family satisfying the assumptions of Theorem 1.3. Let us compare the estimate given in Theorem 1.3 with previously known results depending on the assumptions on Λ.

• If, in addition, Λ satisfies the classical gap condition i.e. there exists γ > 0 such that |λ -µ| > γ for any λ, µ ∈ Λ, λ = µ, then we have c γ λ = 0 (see (1.19)). It follows that (1.21) reduces to

q λ L 2 (0,T ) ≤ C T,α, N exp C α, N (1 + log(λ/γ))λ α .
This estimate is more accurate than the one given in [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF]. On the contrary, because of the log(λ) term in the exponential, this estimate is slightly suboptimal with respect to the one given in [START_REF] Benabdallah | Sharp estimates of the onedimensional boundary control cost for parabolic systems and application to the $n$dimensional boundary null controllability in cylindrical domains[END_REF]. In Remark 3.1, under a more accurate assumption on the asymptotic behaviour of N than (1.20), we drop this log term and thus recover the estimates of [START_REF] Benabdallah | Sharp estimates of the onedimensional boundary control cost for parabolic systems and application to the $n$dimensional boundary null controllability in cylindrical domains[END_REF]. For detailed estimates in the presence of a gap condition we refer to [START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF].

• If we assume now that Λ satisfies a weak gap condition i.e. there exists γ > 0 and p ∈ N * such that

# Λ ∩ [µ, µ + γ] ≤ p, ∀µ > 0,
then we have c γ λ ≤ p -1. With such weak gap condition, uniform estimates (with respect to different parameters) are given in [START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF] as a byproduct of the block moments method. We also mention the recent paper [START_REF] Burgos | Sharp estimates of the one-dimensional boundary control cost for parabolic systems[END_REF] where precise estimates, including time-dependency of the constants, are given in this setting. Let us also mention that estimates for biorthogonal families with a weak gap condition (with p = 2) were already obtained in [START_REF] Cîndea | Particle supported control of a fluid-particle system[END_REF]. In their work, they uses an extra assumption stating that the gap between two eigenvalues should not go to zero too quickly.

• In the absence of any (weak or strong) gap-type condition, the only previous result is [START_REF] Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF] where the biorthogonal family is estimated using the condensation index. In the case where the family Λ satisfies a weak gap condition with p = 2 i.e. there exists γ > 0 such that

# Λ ∩ [µ, µ + γ] ≤ 2, ∀µ > 0,
we recover in (1.21) the same behavior when λ → +∞ as the one given in [START_REF] Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF][START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF][START_REF] Burgos | Sharp estimates of the one-dimensional boundary control cost for parabolic systems[END_REF]. In more general situations, the estimate given in Theorem 1.3 is sub-optimal. However, with the spectral analysis of Theorem 1.2, let us notice that the estimates given in [START_REF] Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF][START_REF] Benabdallah | A block moment method to handle spectral condensation phenomenon in parabolic control problems[END_REF][START_REF] Burgos | Sharp estimates of the one-dimensional boundary control cost for parabolic systems[END_REF] would not allow to prove Theorem 1.1. In fact, we will deal with the sequence of families Λ n which satisfies, for each n, a weak gap condition but with a parameter p going to +∞ with n. Thus, it is crucial to notice that Theorem 1.3 do not require any gap-type condition and that the constants appearing in estimate (1.21) only depend on the asymptotic of the counting function.

To be completely accurate, let us precise that most of the above mentioned results apply in the slightly more general situation where the sequence Λ is only assumed to satisfy λ∈Λ 1 λ < +∞, without any other assumption on the counting function. Our assumption (1.20) is a little bit more restrictive but the absence of any gap-type condition and the uniformity of the estimates offer a new variety of applications.

Results on the minimal time of null controllability of (1.1).

As already stated, one of the achievements of this article is to provide an alternative proof of Theorem 1.1 using the moments method. Let us roughly explain how the results presented in Sections 1.4.1 and 1.4.2 will eventually lead to Theorem 1.1.

As already mentioned, the new term in the estimate given in Theorem 1.3 is

λ+γ ρ λ c γ λ .
In our application to Grushin equation (1.1), for eigenvalues λ in the intermediate range where the uniform gap property is not proved, the size of this term will rely on the following two facts:

• from Proposition 1.2, the term ρ λ is of order exp -√ λ ;

• from (1.15) and (1.13), we have c γ λ which is of order (1 -τ ) √ λ.

Thus, we will obtain a biorthogonal element of order exp (1 -τ )λ + O( √ λ log λ) . Choosing τ sufficiently close to 1, the contribution of this biorthogonal element in the construction of the control (1.6) will be negligible compared to the dissipation. The need of a sufficiently large time will thus simply come from the insufficient observation of the eigenfunctions stated in Proposition 1.1. For more details, we refer to Section 4 and particularly estimate (4.4).

One of the main advantage of the machinery developed through the moments method is that Theorem 1.1 extends to cascade systems with one single control at no cost.

Let d ≥ 2 and consider the controlled cascade system of d Grushin equations

     ∂ t F (t, x, y) + (DI d + K d )F (t, x, y) = B d 1 ω u(t, x, y), (t, x, y) ∈ (0, T ) × Ω, F (t, x, y) = 0, (t, x, y) ∈ (0, T ) × ∂Ω, F (0, x, y) = F 0 (x, y) ∈ (L 2 (Ω)) d , (x, y) ∈ Ω, ( 1
B d :=        1 0 0        , K d :=        0 0 0 0 1 1 0 0        .
We then obtain the following result Theorem

1.4. Let 0 < a < b ≤ 1. The minimal null control time of (1.22) with ω = (a, b) × (0, 1) is T * = a 2 2 .
This theorem is proved in Section 4.2. Notice that the first component of (1.22) is nothing but the controlled scalar Grushin equation (1.1). Thus, the lack of null controllability in time T ≤ a 2 2 follows directly from [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF]. Moreover, our strategy also allows to consider the following cascade system of d Grushin equations with a single boundary control

         ∂ t F (t, x, y) + (DI d + K d )F (t, x, y) = 0, (t, x, y) ∈ (0, T ) × Ω, F (t, x, y) = 0, (t, x, y) ∈ (0, T ) × (∂Ω\Γ 1 ), F (t, x, y) = B d u(t, y), (t, x, y) ∈ (0, T ) × Γ 1 , F (0, x, y) = F 0 (x, y) ∈ (L 2 (Ω)) d , (x, y) ∈ Ω, (1.23)
where Γ 1 has been defined by Γ 1 = {1} × [0, 1]. The functional space considered here for the boundary control is L 2 ((0, T ) × (0, 1)). We then obtain the following result Theorem 1.5. The minimal null control time of (1.23) is

T * = 1 2 .
This theorem is proved in Section 4.3. Contrarily to the scalar case, this result cannot be deduced from Theorem 1.4 using extensions and cut-off functions. It also seems to be out of reach using the Carleman inequalities as in [START_REF] Beauchard | Minimal time issues for the observability of Grushintype equations[END_REF].

Spectral analysis of operator A n .

In this section, we prove respectively Theorem 1.2 in L τ,n , Theorem 1.2 in U n and Proposition 1.2.

Study of the gap property.

To deal with low frequencies it is fundamental to notice that the operator A n is somehow related to the quantum harmonic oscillator. This idea was already present in [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF]. Indeed, for any n ∈ N * , let Ω n := (-√ nπ, √ nπ) and define the following operator

A n := -∂ xx + x 2 , D( A n ) := H 2 (Ω n ) ∩ H 1 0 (Ω n ). (2.1)
Introducing the following change of variables

v k,n (x) := 1 (nπ) 1 4 g k,n x √ nπ , ∀x ∈ Ω n , ∀k ∈ N * , (2.2)
it turns out that v k,n are eigenvectors of the truncated harmonic oscillator A n with associated eigenvalue

λ k,n nπ i.e. A n v k,n = λ k,n nπ v k,n . (2.3)
Thus it is reasonable to expect, at least for large values of n, to obtain valuable informations from the quantum harmonic oscillator on the real line defined by

A := -∂ xx + x 2 , D(A) := G ∈ H 2 (R) ; x → xG(x) ∈ L 2 (R) ,
and more precisely, on its eigenelements which are explicitly given by (see for instance [START_REF] Szegó | Orthogonal polynomials[END_REF])

µ k = 2k -1, G k (x) = 1 π 1/4 2 k-1 2 (k -1)! e -x 2 2 H k (x), ∀k ∈ N * ,
where H k is the (k -1) th Hermite polynomial. Let us notice that since the potential x → x 2 is even, the following Lemma holds.

Lemma 2.1. Let n ∈ N * . • If k is even, then g k,n ,v k,n and G k are odd functions. • If k is odd, then g k,n ,v k,n and G k are even functions.
The comparison between A and A n is at the heart of Section 2.1.2 where we prove the uniform gap-property for low frequencies. It will also allow us to obtain, in Section 2.1.1, a general lower bound on λ k,n that will be used all along this article. The regime of frequencies in which this comparison is relevant is exactly the one given by L τ,n where, in some sense, n is large with respect to k (see Figure 1.2).

The gap property for high frequencies is proved in Section 2.1.3.

A general lower bound on eigenvalues.

We start proving an inequality, which is, unlike Theorem 1.2, valid for the whole spectrum of A n .

Proposition 2.1. The following lower bound holds

λ k,n ≥ nπ(2k -1), ∀k, n ∈ N * .
Proof. The proof is a direct consequence of the min-max principle. Let k ∈ N * and consider the form domain of the operator A i.e.

D := G ∈ H 1 (R) ; x → xG(x) ∈ L 2 (R) .
We have

µ k = min V k ⊂D dim V k =k max G∈V k R G (x) 2 + x 2 G(x) 2 dx G 2 L 2 (R)
.

For any n ∈ N * , let us consider

V k,n := span {v 1,n , . . . , v k,n } ,
as a subspace of L 2 (R) where the functions v j,n are extended by 0 outside Ω n . Notice that,

V k,n ⊂ D, and dim V k,n = k.
Thus,

µ k ≤ max G∈V k,n G L 2 (R) =1 R G (x) 2 +x 2 G(x) 2 dx = max G∈V k,n G L 2 (Ωn ) =1 Ωn G (x) 2 +x 2 G(x) 2 dx = λ k,n nπ ,
which ends the proof.

Uniform gap property for low frequencies.

We start dealing with low frequencies. As we will compare the truncated harmonic oscillator on Ω n with the harmonic oscillator A we will need to prove that, if k is not too large compared to n, the k-th eigenfunction G k of A is small outside the interval Ω n = (-√ nπ, √ nπ). We can precisely state the following two results, whose proofs are postponed to Appendix (see respectively Section 5.2 and Section 5.1).

Lemma 2.2. There exists an integer

n 0 ∈ N * such that Ωn G 2 k (x)dx ≥ 1 2 , ∀k ≤ π 2 n, ∀n ≥ n 0 . Lemma 2.3. Let τ ∈ (0, 1). There exist n 0 (τ ) ∈ N * , C 1 (τ ), C 2 (τ ) ∈ R * + such that 0 ≤ G k (x) ≤ C 1 e -C2n , ∀x ≥ √ nπ, ∀k ≤ τ n π 2 , ∀n ≥ n 0 . (2.4)
The rest of this section is dedicated to the proof of the following proposition. Proposition 2.2. Let τ ∈ (0, 1). There exist

n 0 (τ ) ∈ N * , c 1 (τ ), c 2 (τ ) ∈ R * + , such that λ k,n nπ ≤ µ k + c 1 e -c2n , ∀k ∈ L τ,n , ∀n ≥ n 0 . (2.5)
Together with Proposition 2.1, this states that for low frequencies, the rescaled eigenvalues of the truncated harmonic oscillator are exponentially close (with respect to the parameter n) to the eigenvalues of the harmonic oscillator A. As µ k = 2k -1, we immediately get the following corollary.

Corollary 2.1. Let τ ∈ (0, 1). There exist n 0 (τ ) ∈ N * such that we have the gap property

λ k+1,n -λ k,n ≥ 1, ∀k ∈ L τ,n , ∀n ≥ n 0 .
(2.6)

Proof (of Proposition 2.2). We fix τ ∈ (0, 1). Let n 0 (τ ) be such that Lemmas 2.2 and 2.3 hold. We fix n ≥ n 0 and we are going to prove (2.5) by a strong induction on k.

The case k = 1 is already proved in [6, p. 83]. More precisely, those authors obtained

λ 1,n ≤ nπ + O(n 9/4 e -nπ/2 ),
which is indeed the desired bound. We will actually adapt the proof of [START_REF] Beauchard | Null controllability of Grushin-type operators in dimension two[END_REF] to get the general case.

Suppose now that the result is proved up to rank k -

1 ≥ 1. • Case 1. k is odd In that case, G k is even. Let us define G k = (G k -G k ( √ nπ)) - k-1 p=1 v p,n G k -G k ( √ nπ), v p,n L 2 (Ωn) .
Notice that, by construction, we have

G k ∈ span(v 1,n , v 2,n , . . . , v k-1,n ) ⊥ and G k (± √ nπ) = 0. Thus, G k ∈ H 1 0 (Ω n
) and from the max-min principle, it comes that

λ k,n nπ ≤ A n G k , G k L 2 (Ωn) G k 2 L 2 (Ωn)
.

(2.7)

Let us now compute the image of G k by the operator A n .

A n G k = -G k + x 2 G k -x 2 G k ( √ nπ) - k-1 p=1 (-v p,n + x 2 v p,n ) G k -G k ( √ nπ), v p,n L 2 (Ωn) =µ k G k -x 2 G k ( √ nπ) - k-1 p=1 λ p,n nπ v p,n G k -G k ( √ nπ), v p,n L 2 (Ωn) .
Then, using that G k ⊥ v p,n for any p ∈ {1, ..., k -1}, and again the definition of G k , we find

A G k , G k L 2 (Ωn) = Ωn µ k G k (x) G k (x) -x 2 G k ( √ nπ) G k (x)dx = Ωn µ k G 2 k (x) + G k ( √ nπ)(µ k -x 2 ) G k (x)dx.
This gives the following expression for the Rayleigh quotient we are interested in

A n G k , G k L 2 (Ωn) G k 2 L 2 (Ωn) = µ k + G k ( √ nπ) Ωn (µ k -x 2 ) G k (x)dx G k 2 L 2 (Ωn)
.

(2.8) By (2.7) and the Cauchy-Schwarz inequality, we get

λ k,n nπ ≤ µ k + G k ( √ nπ) µ k -x 2 L 2 (Ωn) G k L 2 (Ωn) .
(2.9)

Note that G k ( √ nπ) is non negative as shown in Lemma 2.3. We now estimate the different terms in the previous right-hand side. The behavior of G k ( √ nπ) is given by Lemma 2.3. Since µ k = (2k -1) ≤ nπ, there exists C > 0 independent of n and k (for instance, C = (2 + 2 5 )π 5/2 ) such that:

µ k -x 2 2 L 2 (Ωn) ≤ C n 5/2 .
(2.10)

Let us now estimate G k L 2 (Ωn) . Thanks to Parseval equality and the fact that

G k ∈ span(v 1,n , v 2,n , . . . , v k-1,n ) ⊥ , we have Ωn G 2 k (x)dx = p≥1 ( G k , v p,n ) 2 L 2 (Ωn) = p≥k ( G k , v p,n ) 2 L 2 (Ωn) . Now, since (v k,n ) k≥1 are pairwise orthogonal, Ωn G 2 k (x)dx = p≥k (G k -G k ( √ nπ), v p,n ) 2 L 2 (Ωn) , (2.11)
and thus, using again Parseval equality,

Ωn G 2 k (x)dx = Ωn G k (x) -G( √ nπ) 2 dx- p<k (G k -G k ( √ nπ), v p,n ) 2 L 2 (Ωn) .
(2.12)

Let us now look at (G k -G k ( √ nπ), v p,n ) 2 L 2 (Ωn) . We have G k -G k ( √ nπ), v p,n L 2 (Ωn) λ p,n nπ = G k -G k ( √ nπ), A n v p,n L 2 (Ωn) = A n G k -x 2 G k ( √ nπ), v p,n L 2 (Ωn) = µ k G k -x 2 G k ( √ nπ), v p,n L 2 (Ωn) =µ k G k -G k ( √ nπ), v p,n L 2 (Ωn) + G k ( √ nπ) µ k -x 2 , v p,n L 2 (Ωn) .
Thus,

G k -G k ( √ nπ), v p,n L 2 (Ωn) λ p,n nπ -µ k = G k ( √ nπ) µ k -x 2 , v p,n L 2 (Ωn)
and,

G k -G k ( √ nπ), v p,n 2
L 2 (Ωn) ≤ G k ( √ nπ) 2 µ k -x 2 2 L 2 (Ωn) λp,n nπ -µ k 2 .
(2.13) Let p < k. By the induction hypothesis,

λ p,n nπ ≤ µ p + c 1 e -c2n .
As c 1 and c 2 only depend on τ , we can increase n 0 (only depending on τ ) such that

c 1 e -c2n ≤ 1, ∀n ≥ n 0 (τ ). (2.14)
Therefore,

µ k - λ p,n nπ ≥ µ k -(µ p + 1) ≥ 2(k -p) -1 > 0.
Thus, with (2.13), we obtain

p<k (G k -G k ( √ nπ), v p,n ) 2 L 2 (Ωn) ≤ π 2 6 G k ( √ nπ) 2 µ k -x 2 2 L 2 (Ωn) . (2.15)
Using (2.10) and plugging it in (2.12) it comes that

Ωn G 2 k (x)dx ≥ Ωn (G k (x) -G k ( √ nπ)) 2 dx - π 2 6 C n 5/2 G k ( √ nπ) 2 . (2.16)
We now estimate Ωn (G k (x) -G k ( √ nπ)) 2 dx thanks to Lemma 2.2. Indeed, as n ≥ n 0 , we have

Ωn (G k (x) -G k ( √ nπ)) 2 dx ≥ 1 2 Ωn G 2 k (x)dx -2 √ nπG 2 k ( √ nπ) ≥ 1 4 -2 √ nπG 2 k ( √ nπ).
Going back to (2.16), we obtain

Ωn G 2 k (x)dx ≥ 1 4 -G k ( √ nπ) 2 2 √ nπ + π 2 6 C n 5/2 .
(2.17)

Finally, using (2.4), for n sufficiently large, depending only on τ , we obtain

Ωn G 2 k (x)dx ≥ 1 8 . (2.18)
Then, using (2.10), (2.18) in (2.9) leads to

λ k,n nπ ≤ µ k + √ 8C n 5/4 G k ( √ nπ). (2.19)
The estimate of Lemma 2.3 proves (2.5) when k is odd.

• Case 2. k is even.

In that case, G k is odd and we set now:

G k = G k - x √ nπ G k ( √ nπ) - k-1 p=1 v p,n G k - x √ nπ G k ( √ nπ), v p,n L 2 (Ωn)
.

Inequality (2.8) becomes:

Ωn ( G k (x)) 2 + x 2 G 2 k (x)dx G k 2 L 2 (Ωn) = µ k + G k ( √ nπ) Ωn µ k - x 3 √ nπ G k (x)dx G k 2 L 2 (Ωn)
.

Inequality (2.10) is replaced by

µ k - x 3 √ nπ 2 L 2 (Ωn) ≤ C n 5/2 ,
with C = (2 + 2 7 )π 5/2 and we conclude in the exact same way.

Uniform gap property for high frequencies.

The goal of this section is to prove the following theorem: Theorem 2.1. We have the lower bound

λ k+1,n -λ k,n ≥ π 1 + π , (2.20)
for any k, n ≥ 1 for which λ k,n satisfies

λ k,n ≥ n 2 π 2 + nπ, (2.21
(λ k,n , g k,n ) -∂ xx g k,n (x) + (nπx) 2 g k,n (x) = λ k,n g k,n (x), for x ∈ (-1, 1), g k,n (±1) = 0, (2.22)
and we assume, possibly changing the sign of g k,n , that ∂ x g k,n (-1) > 0.

We use the idea of [START_REF] Pöschel | Inverse spectral theory[END_REF], where the authors replace the eigenvalue λ k,n by a new variable, denoted here by λ. Hence, for any λ ∈ R, we consider the following Cauchy problem:

     -∂ xx h n (λ, x) + (nπx) 2 h n (λ, x) = λh n (λ, x), for x ∈ (-1, 1), h n (λ, -1) = 0, ∂ x h n (λ, -1) is positive and chosen so that h n (λ, •) L 2 (-1,1) = 1.
(2.23)

The modified Prüfer's change of variable consists in introducing two functions ρ n > 0 and θ n satisfying :

h n (λ, x) = ρ n (λ, x) cos(θ n (λ, x))(λ -n 2 π 2 x 2 ) -1/4 , ∂ x h n (λ, x) = ρ n (λ, x) sin(θ n (λ, x))(λ -n 2 π 2 x 2 ) 1/4 .
(2. [START_REF] Harris | Asymptotics of eigenvalues for regular sturm-liouville problems[END_REF] We impose that the variable λ satisfies λ > n 2 π 2 , so that the expressions in (2.24) make sense. Under this condition, existence of θ n and ρ n is a classical result. Notice that, since h n (λ, -1) = 0 and ∂ x h n (λ, -1) is positive, we can impose without loss of generality

θ n (λ, -1) = π 2 ,
and in that case θ n and ρ n are uniquely determined. Moreover, one can show (see [START_REF] Birkhoff | Introductions to higher mathematics[END_REF] or [START_REF] Harris | Asymptotics of eigenvalues for regular sturm-liouville problems[END_REF]) that θ n is the unique solution of the following Cauchy problem

     ∂ x θ n (λ, x) = -λ -n 2 π 2 x 2 + n 2 π 2 x sin(2θ n (λ, x)) 2(λ -n 2 π 2 x 2 ) , for x ∈ [-1, 1], θ n (λ, -1) = π 2 (2.25)
Let us start with the following lemma which is given without proof in [22, Lemma 2].

Lemma 2.4. Let λ k,n be an eigenvalue such that λ k,n ≥ n 2 π 2 + nπ. The solution θ n of (2.25) satisfies

θ n (λ k,n , 1) = π 2 -kπ. Proof. First, notice that h n (λ k,n , x) = g k,n (x)
. According to Sturm oscillation Theorem, the function x ∈ [-1, 1] → h n (λ k,n , x) has exactly k + 1 zeros. Thus, given that ρ n does not vanishes in [-1, 1], there exists exactly k + 1 points

-1 = x 0 < x 1 < • • • < x k = 1 such that ∀i ∈ 0, k , θ n (λ k,n , x i ) = π/2 mod π. Since θ n (λ k,n , x 0 ) = π 2 , by continuity of x → θ n (λ k,n , x) we have θ n (λ k,n , x 1 ) ∈ { π 2 -π, π 2 , π 2 + π}. But according to (2.25), ∂ x θ n (λ k,n , x 0 ) = -λ -n 2 π 2 x 2 0 < 0 thus θ n (λ k,n , x 1 ) = π 2 + π, otherwise, by continuity of x → θ n (λ k,n , x) there would exist a point x ∈ (x 0 , x 1 ) such that θ n (λ k,n , x) = π/2. Since also ∂ x θ n (λ k,n , x 1 ) < 0, θ n (λ k,n , x 1 ) = π 2 , hence θ n (λ k,n , x 1 ) = π 2 -π.
We deduce with an immediate induction argument that θ n (λ k,n , x k ) = π 2 -kπ. According to Lemma 2.4 and the fact that λ → θ n (λ, 1) is continously differentiable (see [30, Theorem 2.5.2]), we have

θ n (λ k+1,n , 1) -θ n (λ k,n , 1) = -π = λ k+1,n λ k,n ∂ λ θ n (λ, 1)dλ. It follows that π ≤ (λ k+1,n -λ k,n ) sup λ∈[λ k,n ,λ k+1,n ] |∂ λ θ n (λ, 1)|, (2.26)
We shall now estimate this supremum. For this purpose, let's compute the derivative with respect to λ of (2.25):

∂ λ (∂ x θ n ) = - 1 2 √ λ -n 2 π 2 x 2 - n 2 π 2 x 2(λ -n 2 π 2 x 2 ) 2 sin(2θ n )+ n 2 π 2 x cos(2θ n ) λ -n 2 π 2 x 2 ∂ λ θ n (2.27)
by regularity of θ n , we can exchange the derivatives and use that ∂ λ θ n (λ, -1) = 0 to get, for any x ∈ (-1, 1),

∂ λ θ n (λ, x) = x -1 - 1 2 λ -n 2 π 2 y 2 - n 2 π 2 y 2(λ -n 2 π 2 y 2 ) 2 sin(2θ n (λ, y)) + n 2 π 2 y cos(2θ n (λ, y)) λ -n 2 π 2 y 2 ∂ λ θ n (λ, y) dy, so that |∂ λ θ n (λ, x)| ≤ x -1 1 2 λ -n 2 π 2 y 2 + n 2 π 2 |y| 2(λ -n 2 π 2 y 2 ) 2 + n 2 π 2 |y| λ -n 2 π 2 y 2 |∂ λ θ n (λ, y)| dy.
Applying Gronwall's Lemma, we obtain

|∂ λ θ n (λ, 1)| ≤ 1 -1 1 2 λ -n 2 π 2 y 2 exp 1 y n 2 π 2 |s| λ -n 2 π 2 s 2 ds dy + 1 -1 n 2 π 2 |y| 2(λ -n 2 π 2 y 2 ) 2 exp 1 y n 2 π 2 |s| λ -n 2 π 2 s 2 ds dy,
and finally, using the fact that the functions involved are even,

|∂ λ θ n (λ, 1)| ≤ 1 0 1 λ -n 2 π 2 y 2 dy + 1 0 n 2 π 2 y (λ -n 2 π 2 y 2 ) 2 dy × exp 1 0 2n 2 π 2 s λ -n 2 π 2 s 2 ds .
(2.28)

The three integrals appearing in the right-hand side can be bounded explicitely (we recall that λ > n 2 π 2 + nπ) as follows

1 0 1 λ -n 2 π 2 y 2 dy = 1 nπ nπ √ λ 0 dt √ 1 -t 2 ≤ 1 2n , 1 0 n 2 π 2 y (λ -n 2 π 2 y 2 ) 2 dy = 1 2 1 (λ -n 2 π 2 y 2 ) 1 0 ≤ 1 2(λ -n 2 π 2 ) ≤ 1 2nπ
,

and 1 0 2n 2 π 2 s λ -n 2 π 2 s 2 ds = -log(λ -n 2 π 2 s 2 ) 1 0 = log λ λ -n 2 π 2 ≤ log(1 + nπ).
Gathering those estimates in (2.28), we finally get

|∂ λ θ n (λ, 1)| ≤ 1 2n + 1 nπ e log(1+nπ) ≤ 1 + nπ n ≤ 1 + π.
Injecting this upper bound in (2.26), we get

π ≤ (λ k+1,n -λ k,n )(1 + π),
and the claim is proved. Remark 2.2. In [1, Theorem 4.5.1], it is proved that for any ε > 1 exp(π)-1 > 0, there exist C ε > 0 and n 0 ∈ N * both depending only on ε such that

λ k+1,n -λ k,n ≥ C ε λ k,n , ∀n ≥ n 0 , ∀k ≥ π 2 n(1 + ε).
The range of application of this estimate is slightly more restrictive than U n but gives a lower bound for the gap that goes to infinity with k. This is coherent with the linear behavior observed on the gap in Figure 1.2 for high frequencies and with the general theory of Sturm Liouville operators stating that, for a fixed n ∈ N * , one recovers an asymptotic quadratic behavior for the eigenvalues (see for instance [2, Theorem 1.1]).

Conclusion.

We have proved two gap properties, one for the eigenvalues whose indices lie in L τ,n and another one for the eigenvalues whose indices belong to U n . To conclude, let us show that the gap property holds for the union: L τ,n ∪ U n . Take k 1 ∈ L τ,n and k 2 ∈ U n . According to Proposition 2.2 and estimate (2.14),

λ k1,n nπ ≤ (2k 1 -1) + 1 = 2k 1 .
Hence, applying Proposition 2.1 with

k 2 ∈ U n , λ k2,n -λ k1,n ≥ nπ 2(k 2 -k 1 ) -1) ≥ nπ.

Non-uniform gap estimate for middle frequencies.

This section is dedicated to the proof of Proposition 1.2. From Proposition 2.1, as τ ≥ 3 4 , we have

k ≥ τ n π 2 =⇒ λ k,n ≥ nπ 2τ n π 2 -1 ≥ 3 4 n 2 π 2 -nπ,
which implies (1.13).

The estimate (1.14) will follow adapting the arguments of [2, Theorem 1.1]. Let λ and λ in Λ n . Let u and ũ the associated normalized eigenfunctions. Setting w(x) := u (-1)ũ(x) -ũ (-1)u(x), we obtain w(-1) = w (-1) = 0 and we obtain

-w (x) + n 2 π 2 x 2 w(x) = λw + ( λ -λ)u (-1)ũ(x).
W (x) ≤ exp x -1 n 2 π 2 s 2 √ λ ds x -1 λ -λ √ λ u (-1)ũ(s) ds ≤ √ 2 exp 2 3 n 2 π 2 √ λ | λ -λ| √ λ |u (-1)|.
Noticing that the two eigenfunctions are normalized and orthogonal, thus

|u (-1)| 2 ≤ 1 -1 |u (-1)ũ(s) -ũ (-1)u(s)| 2 ds ≤ 1 -1 W (s) 2 ds ≤ 2 √ 2 exp 2 3 n 2 π 2 √ λ | λ -λ| √ λ 2 |u (-1)| 2 .
As u (-1) = 0, it comes that

|λ -λ| ≥ √ λ 2 exp - 2 3 n 2 π 2 √ λ .
Let k ≥ τ n π 2 and j ≥ 1, j = k. We apply this estimate to λ = λ k,n and λ = λ j,n . Taking into account (1.13), we obtain

|λ k,n -λ j,n | ≥ λ k,n 2 exp - 2 3 4λ k,n λ k,n .
which ends the proof of Proposition 1.2.

The biorthogonal family.

The goal of this section is to prove the existence of a biorthogonal family to the exponentials as well as the suitable estimates stated in Theorem 1.3. We start with some technical estimates on the inverse of Blaschke-type products and their derivatives in the considered setting. We use these estimates in Section 3.2 to prove Theorem 1.3 and finally extend it to biorthogonal families to t → (-t) p e -λt ; p ∈ 0, d , λ ∈ Λ , in Theorem 3.2.

Preliminary estimates.

We use in this section the notation introduced in Section 1.4.2, in particular (1.16), (1.17) and (1.18).

For any countable family Λ ⊂ (0, +∞) such that λ∈Λ 1 λ < +∞ and any λ ∈ Λ, we define the Blaschke product

W Λ λ (z) := σ∈Λ σ =λ σ -z σ + z , ∀z ∈ C,
which is well-defined and holomorphic on C due to the summability assumption on 1 λ λ∈Λ .

Estimates of Blaschke product.

We start with the following estimate of the reciprocal of the Blaschke product in the neighborhood of λ.

Proposition 3.1. Let α ∈ (0, 1) and N > 0.

• Let Λ be a family such that its counting function N satisfies (1.20).

For any λ ∈ Λ, we have the estimate

1 W Λ λ (x) ≤ x + 2γ ρ λ c γ λ exp C α, N (1 + log (x/γ)) x α , (3.1)
∀γ > 0, ∀x ∈ R, s.t. |x -λ| ≤ min(γ, ρ λ ) 2 ,
where C α, N only depends on the parameters α and N . • Let Λ be a family such that its counting function N satisfies the more precise asymptotic

|N (r) -N r α | ≤ C, ∀r > 0. (3.2)
For any λ ∈ Λ, we have the estimate

1 W Λ λ (x) ≤ x + 2γ ρ λ c γ λ exp C α, N (1 -log (γ)) x α , (3.3) ∀γ > 0, ∀x ∈ R, s.t. |x -λ| ≤ min(γ, ρ λ ) 2 ,
where C α, N only depends on the parameters α and N . Remark 3.1. The difference between the two estimates given in the previous proposition is that, under the stronger assumption (3.2), there is no log x term in the exponential.

Thanks to this slight improvement, the present strategy developed in the rest of the article let us exactly recover the estimates of biorthogonal families under a gap assumption as given in [9, Theorem 1.5] with constants depending only on α and N and on the spectral gap ρ.

However, for our purpose, we focus on the weaker assumption (1.20). Thus, in the following sections we will not consider both settings.

Before proving Proposition 3.1, let us recall some useful formulas making use of the counting function. These results are straightforward and we omit the proofs. For detailed proofs we refer to [START_REF] Boyer | Controllability of linear parabolic equations and systems[END_REF].

Lemma 3.1. Let f : (0, +∞) → R be of class C 1 .

• For any r > 0,

λ∈Λ λ≤r f (λ) = f (r)N (r) - r 0 f (t)N (t)dt.
• For any 0 ≤ s < r,

λ∈Λ s<λ≤r f (λ) = f (r)N (r) -f (s)N (s) - r s f (t)N (t)dt.
• For any r > 0 such that the following quantities are well defined,

λ∈Λ λ>r f (λ) = -f (r)N (r) - +∞ r f (t)N (t)dt.
From now on, we introduce the remainder of the series of the reciprocals of the eigenvalues

R(r) := λ∈Λ λ>r 1 λ ,
and we observe that it is related to the counting function through the formula N (r) -N (s) ≤ rR(s), ∀0 < s < r.

(3.4) Lemma 3.2. Assume that the counting function associated with a family Λ satisfies (1.20) then we have

inf Λ ≥ 1 N 1 α , and
R(r) ≤ N 1 -α r α-1 , ∀r > 0.
We now have all the ingredients to proceed to the proof of Proposition 3.1.

Proof (of Proposition 3.1).

We first observe that, by definition of ρ λ we have

(λ -ρ λ , λ + ρ λ ) ∩ Λ = {λ},
and in particular, since |x -λ| ≤ ρ λ /2, we have (0, x) ∩ Λ = (0, λ) ∩ Λ, and (x, +∞) ∩ Λ = (λ, +∞) ∩ Λ.

(3.5)

We will prove the estimate under the assumption that

x + γ 2 ∈ Λ, (3.6)
the general case being deduced by density. Let us write the Blaschke product as

1 W Λ λ (x) = σ∈Λ σ =λ 1 + x σ σ∈Λ σ =λ 1 -x σ =: Q(x) D(x) .
First step: let us prove the following upper bound on the numerator

Q(x) ≤ exp C α, N x α .
We split Q(x) into two parts

Q 1 (x) := σ∈Λ σ<λ 1 + x σ and Q 2 (x) := σ∈Λ σ>λ 1 + x σ ,
and by using (3.5) we have

Q 1 (x) = σ∈Λ σ<x 1 + x σ and Q 2 (x) = σ∈Λ σ>x 1 + x σ ,
For the first term, we have 1 ≤ x/σ. Thus, from Lemma 3.1 and (1.20), it comes that

log Q 1 (x) ≤ σ∈Λ σ<x log 2x σ = N (x) log 2 + x 0 1 t N (t) dt ≤ N (log 2)x α + N α x α .
For the second term, from Lemma 3.2, we have

Q 2 (x) ≤ σ∈Λ σ>x e x σ = exp   x σ∈Λ σ>x 1 σ    = e xR(x) ≤ exp N 1 -α x α .
Second step: we now prove the following lower bound on D(x)

D(x) ≥ exp -C α, N (1 + log (x/γ)) x α ρ λ 2x + γ c γ λ , if N satisfies (1.20) and D(x) ≥ exp -C α, N (1 -log γ) x α ρ λ 2x + γ c γ λ , if N satisfies (3.2).
To this end, we study the five following terms

D 1 (x) := σ∈Λ σ≤x/2 1 - x σ , D 2 (x) := σ∈Λ x/2<σ≤x-γ/2 1 - x σ , D 3 (x) := σ∈Λ,σ =λ x-γ/2<σ≤x+γ/2 1 - x σ , D 4 (x) := σ∈Λ x+γ/2<σ≤2x 1 - x σ and D 5 (x) := σ∈Λ σ>2x 1 - x σ .
The only difference between assumptions (1.20) and (3.2) will appear in the estimates of D 2 and D 4 .

• For any σ ≤ x 2 , we have 1

-x σ ≥ 1 and thus D 1 (x) ≥ 1. • To treat D 5 (x), we notice that 1 -u ≥ e -2u , ∀u ∈ 0, 1 2 .
Thus,

D 5 (x) ≥ exp   -2x σ∈Λ σ>2x 1 σ    ≥ e -2xR(2x) .
Then, Lemma 3.2 implies

D 5 (x) ≥ exp - N 1 -α (2x) α .
• We now turn to the term D 3 (x). Thanks to (3.6) we have

D 3 (x) := σ∈Λ,σ =λ x-γ/2<σ<x+γ/2 1 - x σ .
Recall that ρ λ and c γ λ are defined by (1.17) and (1.18). In particular, we have |λ -σ| ≥ ρ λ and by assumption |λ -x| ≤ ρ λ /2, so that we finally have

|x -σ| ≥ ρ λ /2, ∀σ ∈ Λ, σ = λ.
It follows that

D 3 (x) ≥ σ∈Λ,σ =λ σ∈(x-γ/2,x+γ/2) ρ λ 2(x + γ/2)
.

There is now two subcases:

-If ρ λ
2(x+γ/2) ≤ 1, then we can use that (x -γ/2, x + γ/2) ⊂ (λ -γ, λ + γ) (because we assumed that |x-λ| ≤ γ/2) to obtain the bound from below

D 3 (x) ≥ σ∈Λ,σ =λ σ∈(λ-γ,λ+γ) ρ λ 2(x + γ/2) = ρ λ 2x + γ c γ λ , ( 3
≥ σ∈Λ x+γ/2<σ≤2x log σ -x 2x = N (2x) log(1/2) -N (x + γ/2) log(γ/4x) - 2x x+γ/2 1 t -x N (t) dt = -(log 2)N (2x) + N (x + γ/2) log(4x/γ) - x γ/2 1 u N (u + x) du.
Notice that,

x γ/2 1 u dt = log 2x γ .
Thus,

log D 4 (x) ≥ -(log 2)N (2x) + (log 2)N (x + γ/2) - x γ/2 1 u (N (u + x) -N (γ/2 + x)) du. (3.8)
The difference between assumptions (1.20) and (3.2) appears in the estimate of

N (u + x) -N (γ/2 + x).
-Assume first, that N satisfies (1.20). Formula (3.4) implies

N (u + x) -N (γ/2 + x) ≤ (u + x)R(γ/2 + x) ≤ 2xR(x), which leads to log D 4 (x) ≥ -(log 2)N (2x) + (log 2)N (x + γ/2) -2xR(x) log(2x/γ).
Finally, from (1.20) and Lemma 3.2, we deduce that

log D 4 (x) ≥ -(log 2) N (2x) α -2 N 1 -α x α log 2x γ
which ends the proof.

-Assume now that N satisfies (3.2). Then,

|N (x + u) -N (x + γ/2)| ≤ 2C + N |(x + u) α -(x + γ/2) α | ≤ 2C + N |u -γ/2| α ≤ 2C + N |u| α .
Integrating this inequality we obtain

x γ/2 1 u (N (x + u) -N (x + γ/2)) du ≤ 2C log 2x γ + N x γ/2 |u| α-1 ≤ 2C log 2x γ + N α x α .
Getting back to (3.8), we obtain

log D 4 (x) ≥ -(log 2)N (2x) -2C log (2x/γ) - N α x α .
Using again (3.2) ends the proof.

Estimates of derivatives of Blaschke product.

We now prove estimates similar to Proposition 3.1 but on derivatives of the inverse of Blaschke products.

Proposition 3.2. Let α ∈ (0, 1) and N > 0. Let Λ be a family such that its counting function N satisfies (1.20).

For any λ ∈ Λ, and any k ≥ 1, we have the estimate

1 W Λ λ (k) (λ) ≤ k! λ + 5 ρ λ c γ λ 2 min(γ, ρ λ ) k exp C α, N 1 + log λ γ λ α , ∀γ ∈ (0, 2),
where C α, N > 0 only depends on the parameters α and N . Remark 3.2. As detailed for Proposition 3.1, the same estimate as the one given in Proposition 3.2 holds without the log(λ) term in the exponential if one replaces assumption (1.20) by (3.2).

Γ λ := z ∈ C, |z -λ| = min(γ, ρ λ ) 2 .
By definition of ρ λ the associated closed disk D λ does not contain any other element of Λ \ {λ} and therefore 1

W Λ λ
is holomorphic in a neighborhood of D λ , and we can apply the Cauchy formula to get

1 W Λ λ (k) (λ) = k! 2iπ Γ λ 1 W Λ λ (z) (z -λ) k+1 dz. It follows that 1 W Λ λ (k) (λ) ≤ k! r k λ sup z∈Γ λ 1 W Λ λ (z)
.

(3.9)

Thus, we are reduced to estimate 1

W Λ λ on Γ λ . For any z ∈ C + := {z ∈ C : (z) > 0}, since |λ -z| ≤ |λ + z|, notice that λ -z λ + z 2 = (λ -z) 2 + ( z) 2 (λ + z) 2 + ( z) 2 ≥ (λ -z) 2 (λ + z) 2 .
Thus, we obtain, for any

z ∈ C + W Λ λ (z) ≥ W Λ λ ( z) .
Notice that, for any λ ∈ Λ, thanks to (1.17), one has λ ≥ ρ λ and thus Γ λ ⊂ C + . From (3.9), we get

1 W Λ λ (k) (λ) ≤ k! r k λ sup x∈[λ-r λ ,λ+r λ ] 1 W Λ λ (x) . (3.10)
By definition of r λ we can apply Proposition 3.1 for every x ∈ [λ -r λ , λ + r λ ]. Then, using that r λ ≤ 1 (since we assumed γ < 2), the claim follows.

3.2. Construction of biorthogonal families.

A biorthogonal family to the exponentials.

In this section, we prove Theorem 1.3. We start with the following construction of a biorthogonal family to the exponentials in infinite time.

Proposition 3.3. Let α ∈ (0, 1), N > 0 and let Λ ⊂ (0, +∞) be a countable family whose counting function N satisfies (1.20).

Then, there exists a family of functions (q λ,∞ ) λ∈Λ ⊂ L 2 (0, +∞) which is biorthogonal to the exponential functions

t → e -λt ; λ ∈ Λ , that is +∞ 0 e -λt q µ,∞ (t)dt = δ λ,µ , ∀λ, µ ∈ Λ,
and that satisfies for any λ ∈ Λ, the estimate

q λ,∞ L 2 (0,+∞) ≤ C α, N λ + γ ρ λ c γ λ exp C α, N (1 + log(λ/γ))λ α , ∀γ > 0, (3.11)
where C α, N only depend on the parameters α and N , and ρ λ , c γ λ are defined in (1.17) and (1.18) respectively.

Remark 3.3. As detailed for Proposition 3.1, the same estimate as (3.11) holds without the log(λ) term in the exponential if one replaces assumption (1.20) by (3.2).

Proof. The proof rely on the isometry induced by the Laplace transform

L : L 2 (0, +∞; C) → H 2 (C + ) f → F : λ → +∞ 0 f (t)e -λt dt
onto the Hardy space H 2 (C + ) endowed with the norm

F 2 H 2 (C + ) = +∞ -∞ |F (iy)| 2 dy.
For more details about this isometry we refer to [28, p. 19-20] and the references therein.

For any given λ ∈ Λ, we define

J λ (z) := 1 + λ W Λ λ (λ) W Λ λ (z) 1 + z , ∀z ∈ C + , which is clearly holomorphic in C + . Moreover, since W Λ λ satisfies |W Λ λ (iy)| = 1, for any y ∈ R, (3.12)
we get that J λ ∈ H 2 (C + ) and that

J λ H 2 (C + ) = +∞ -∞ 1 1 + y 2 dy 1 2 1 + λ |W Λ λ (λ)| = √ π(1 + λ) |W Λ λ (λ)|
.

By Proposition 3.1, we finally obtain the estimate

J λ H 2 (C + ) ≤ C α, N λ + γ ρ λ c γ λ exp C α, N (1 -log λ/γ)λ α .
We define q λ,∞ := L -1 (J λ ) so that, by the isometry property of the Laplace transform, the expected L 2 bound for q λ,∞ holds. Moreover, by the properties of W Λ λ , we have J λ (λ) = 1, and J λ (µ) = 0, ∀µ ∈ Λ\{λ}. (3.13) This translates, thanks to the Laplace transform definition, into the expected orthogonality conditions

+∞ 0 q λ,∞ (t)e -µt dt = δ λ,µ , µ ∈ Λ.
The proof is complete.

For T ∈ (0, +∞], let us now introduce the spaces

A(Λ, T ) = Span t → e -λt , λ ∈ Λ L 2 (0,T )
.

We will use the following general result that is a consequence of the study of Remeztype inequalities in Müntz spaces that can be found for instance in [START_REF] Borwein | Polynomials and Polynomial Inequalities[END_REF][START_REF] Borwein | Generalizations of Müntz's theorem via a Remez-type inequality for Müntz spaces[END_REF]. A detailed proof is proposed in [START_REF] Boyer | Controllability of linear parabolic equations and systems[END_REF]. Theorem 3.1. Let α ∈ (0, 1), N > 0 and Λ ⊂ (0, +∞) be a family whose counting function satisfies (1.20). For any T > 0, there exists a C α, N ,T > 0 depending only on α, N and T such that

f L 2 (0,+∞) ≤ C α, N ,T f L 2 (0,T ) , ∀f ∈ A(Λ, ∞).
The invertibility of the restriction map on A(Λ, ∞) was already proven for instance in [START_REF] Schwartz | Étude des sommes d'exponentielles réelles[END_REF] and uniform estimates (with respect to different parameters) for the above constant were proved in [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF]Theorem 1.3] under an additional gap condition. Here, we need uniform bounds with respect to the counting function and we do not require any gap condition. This is why we will use the estimate given in Theorem 3.1.

We now have all the ingredients to prove Theorem 1.3. Proof (of Theorem 1.3). Let (q λ,∞ ) λ∈Λ be the biorthogonal family to the exponentials in L 2 (0, +∞) given by Proposition 3.3. We start by projecting those functions onto the closed subspace A(Λ, ∞) by setting qλ := Proj A(Λ,∞) q λ,∞ , ∀λ ∈ Λ. By construction we have, for any µ ∈ Λ, +∞ 0 qλ (t)e -µt dt = +∞ 0 q λ,∞ (t)e -µt dt = δ λ,µ , and moreover qλ L 2 (0,∞) ≤ q λ,∞ L 2 (0,∞) . Therefore (q λ ) λ is also a biorthogonal family satisfying the same bounds as (q λ,∞ ) λ but it has the additional property to be included in the set A(Λ, ∞).

For a finite given T > 0, let us define the restriction operator

R Λ,T : f ∈ A(Λ, ∞) → f |(0,T ) ∈ A(Λ, T ).
By Theorem 3.1, we know that it is invertible and that the following uniform bound holds

R -1 Λ,T ≤ C α, N ,T ,
therefore the adjoint (R -1 Λ,T ) * is an operator from A(Λ, ∞) onto A(Λ, T ) that satisfies the same bound.

Then, setting

q λ := (R -1 Λ,T ) * qλ , ∀λ ∈ Λ,
we see that the biorthogonality property in L 2 (0, T ) is guaranteed as well as the expected bounds.

A more general biorthogonal family.

The previous construction can be extended to prove the existence and suitable estimates of a biorthogonal family in L 2 (0, T ) to t → (-t) j e -λt ; j ∈ 0, d , λ ∈ Λ , as detailed in the following theorem. Theorem 3.2. Let α ∈ (0, 1), N > 0, d ∈ N, and let Λ ⊂ (0, +∞) be a countable family whose counting function N satisfies (1.20).

Then, there exists a family of functions (q l λ,T ) l∈ 0,d ,λ∈Λ ⊂ L 2 (0, +∞) which has the following biorthogonality property

T 0 (-t) j e -λt q l µ,T (t)dt = δ λ,µ δ j,l , ∀λ, µ ∈ Λ, ∀j, l ∈ 0, d , (3.14)
and that satisfies for any λ ∈ Λ and j ∈ 0, d , the estimates (3.15) where C α, N ,d and C α, N ,T,d only depend on the parameters α, N , d (and T ). Proof. The proof follows the same line as the one of Theorem 1.3. We only detail how to adapt Proposition 3.3.

q j λ,T L 2 (0,T ) ≤ C α, N ,T,d λ + 5 ρ λ (d+1)c γ λ 2 min(γ, ρ λ ) d+1 × exp C α, N ,d (1 + log(λ/γ))λ α , ∀γ ∈ (0, 2),
For any λ ∈ Λ and any l ∈ 0, d , we are looking for a function

J l λ ∈ H 2 (C + ) satisfying 1. ∀σ ∈ Λ\{λ}, ∀p ∈ 0, d , (J l λ ) (p) (σ) = 0, 2. ∀p ∈ 0, d , (J l λ ) (p) (λ) = δ p,l
, with suitable estimates of its norm. Notice that condition (3.14) amounts to:

L(q l λ ) (p) (σ) = δ λ,σ δ p,l ,
which is given by conditions 1 and 2.

Let us look for J l λ in the following form

J l λ (z) := P l λ (z) W Λ λ (z) β(z) d+1 ,
where β(z) = 1 + z and P l λ is a polynomial of degree d to be determined so that condition 2 is satisfied. Notice that the term W Λ λ d+1 ensures that condition 1 holds, whereas the term 1 β(z) d+1 = 1 (1+z) d makes J l λ integrable on iR as we shall see. Let us now determine the polynomial P l λ . For any k ∈ 0, d , this polynomial should satisfy

P l λ (k) (λ) = J l λ × β W Λ λ d+1 (k) (λ) = k p=0 k p (J l λ ) (p) (λ) β W Λ λ d+1 (k-p) (λ) = k p=0 k p δ p,l β W Λ λ d+1 (k-p) (λ) =        k l β W Λ λ d+1 (k-l) (λ) if k ≥ l, 0 if k < l.
Thus, setting

P l λ (z) = d k=l k l β W Λ λ d+1 (k-l) (λ) (z -λ) k k! (3.16)
ensures condition 2. We now turn to the estimate of J l λ H 2 (C + ) . First, recall that for all y ∈ R, we have W Λ λ (iy) = 1. Therefore,

J l λ 2 H 2 (C + ) = ∞ -∞ |J l λ (iy)| 2 dy = ∞ -∞ P l λ (iy) (1 + iy) d+1 2 dy.
Note that, according to (3.16), there exists C d > 0 such that, for any z ∈ C,

(P l λ )(z) ≤ C d (1 + |z|) d (1 + λ) 2d+1 d-l k=0 1 (W Λ λ ) d+1 (k) (λ) .
Hence,

J l λ H 2 (C + ) ≤ C d (1 + λ) 2d+1 d-l k=0 1 (W Λ λ ) d+1 (k) (λ) . (3.17)
Then, from Leibniz rule and Proposition 3.2, we obtain (3.15). As in the proof of Theorem 1.3, the desired biorthogonal family in L 2 (0, T ; R) will follow from the suitable version of Theorem 3.1 adapted to the new family of functions we consider here, see the details in [START_REF] Boyer | Controllability of linear parabolic equations and systems[END_REF].

Application to cascade systems of Grushin equations.

We gather the spectral analysis done in Section 2 and the general construction of biorthogonal family proved in Section 3 to prove Theorem 1.1. Then, in Section 4.2, we extend these results to the cascade system of coupled Grushin equations (1.22).

Minimal null control time of Grushin equation.

In this section, we prove Theorem 1.1. From the strategy detailed in Section 1, we want to prove that for T > a 2

2 the system (1.4) is null controllable with a control u n uniformly bounded with respect to n. The same arguments as the one of [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF]Proposition 4.1] (see also [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF]Proposition 3.1]) based on cut-off arguments prove that it is sufficient to consider the case ω = (a, 1) × (0, 1).

Recall that from (1.6), the control

u n (t, x) = - k∈N * e -λ k,n T f 0 n , g k,n L 2 (-1,1) g k,n (x) g k,n 2 L 2 (a,1) q k,n (T -t),
formally solves the moments problem (1.5).

To deal with the biorthogonal family (q k,n ) k∈N * , we prove the following uniform bound on the counting function.

Proposition 4.1. There exists C > 0, such that,

λ k,n ≥ Ck 2 , ∀k, n ∈ N * . (4.1)
Thus, there exists C > 0 such that, the counting function N n of the family Λ n (see (1.16)) satisfies

N n (r) ≤ C √ r, ∀r > 0, ∀n ∈ N * .
Proof. In [17, p.104, thm 11], the authors use Rayleigh's Theorem to deduce that

λ k,n = k 2 π 2 4 + r k,n , (4.2)
where |r k,n | ≤ n 2 π 2 for any n, k ∈ N * . To prove (4.1), we consider two different situations • if k ≥ 3n then, according to (4.2), there exists c 1 > 0 such that

λ k,n ≥ k 2 π 2 4 -n 2 π 2 ≥ c 1 k 2 ,
• if k < 3n then, according to Proposition 2.1, there exists c 2 > 0, such that

λ k,n ≥ (2k -1)nπ ≥ kn ≥ c 2 k 2 .
Then we set C := min(c 1 , c 2 ). The uniform asymptotic on N n follows directly. We now get the following bound on the biorthogonal family using Theorem 1.3. The estimate given by Theorem 1.3 will be sufficient for our purpose: in the range of frequencies for which we do not have a uniform gap, we have a poor gap-type inequality (see Proposition 1.2) but which concerns not so much frequencies (see (1.15)).

Proposition 4.2. Let T > 0 and ε > 0. There exists C T,ε > 0 such that, for any n ∈ N * , there exists a biorthogonal family (q k,n ) k∈N * to the exponentials associated with Λ n in L 2 (0, T ; R) i.e.

T 0 e -λ k,n t q j,n (t)dt = δ k,j , ∀k, j ∈ N * , which satisfies q k,n L 2 (0,T ) ≤ C T,ε e ελ k,n , ∀k ∈ N * .

Proof. Let ε > 0. Let τ ∈ 3 4 , 1 be such that

3(1 -τ ) ≤ ε 2 , (4.3)
and γ τ the constant given by Corollary (1.2). Using Theorem 1.3 with the parameter γ τ and Proposition 4.1, there exists such a biorthogonal family which satisfies

q k,n L 2 (0,T ) ≤ C T exp C (1 -log γ τ + log λ k,n ) λ k,n λ k,n + γ τ ρ k,n c γτ k,n
.

We now consider two cases.

• If λ k,n ∈ L τ,n ∪ U n , then, from Corollary 1.2, it comes that ρ k,n ≥ γ τ and thus c γτ k,n = 0. Then, we obtain

q k,n L 2 (0,T ) ≤ C T exp C (1 -log γ τ + log λ k,n ) λ k,n ,
which ends the proof in this case.

• Otherwise, λ k,n ∈ M τ,n . From Proposition 1.2 and Proposition 2.1, we obtain

ρ k,n ≥ λ k,n 2 exp - 8 3 λ k,n ≥ exp -3 λ k,n .
From Corollary 1.2, we have dist(L τ,n , U n ) ≥ γ τ . Together with (1.15) and (1.13), this implies

c γτ k,n ≤ #M τ,n + 1 ≤ (1 -τ ) nπ 2 + 2 ≤ (1 -τ ) λ k,n + 2.
From (4.3), it comes that

1 ρ k,n c γτ k,n ≤ exp 3(1 -τ )λ k,n + 6 λ k,n ≤ exp ε 2 λ k,n + 6 λ k,n . (4.4)
Finally, this implies

q k,n L 2 (0,T ) ≤ C T exp C (1 -log γ τ + log(λ k,n + γ τ )) λ k,n exp ε 2 λ k,n ,
which ends the proof. Remark 4.1. As mentioned in the beginning of Section 1.4.3, estimate (4.4) is the key point to obtain the bound on the biorthogonal family given in Proposition 4.2. The very weak gap information coming from Proposition 1.2 is somehow compensated by the fact that this information does not concern too many eigenvalues (see (1.15)).

We can now prove Theorem 1.1. Proof (of Theorem 1.1). Let T > a 2 2 . Let ε > 0 and η > 0 be such that

T > a 2 2 + η 2 + ε.
From Proposition 1.1, it comes that

g k,n 2
L 2 (a,1) ≥ C η e -nπ(a 2 +η) n 2 π 2 .
From Proposition 2.1, it comes that

λ k,n ≥ nπ(2k -1) ≥ nπ + 2(k -1), ∀k, n ∈ N * .
Together with Proposition 4.2, we obtain that the control u n given by (1.6) satisfies

u n L 2 ((0,T )×(a,1)) ≤ C ε,η k∈N * e -2(k-1)(T -ε) n 2 π 2 e -nπ T -a 2 2 -η 2 -ε f 0 n L 2 (-1,1) ≤ C ε,η,T f 0 n L 2 (-1,1) ,
which ends the proof of Theorem 1.1.

Minimal null control time of cascade systems of Grushin equations.

This section is dedicated to the proof of Theorem 1.4 that is the proof of null controllability in time T > a 2 2 of the cascade system (1.22). As in the scalar case, one of the crucial steps is the following estimate of biorthogonal families.

Proposition 4.3. Let T > 0, ε > 0 and d ≥ 2. There exists C T,ε,d > 0 such that, for any n ∈ N * , there exists a biorthogonal family

(q l k,n ) k∈N * ,l∈ 0,d-1 to t → (-t) l e -λ k,n t ; k ≥ 1, l ∈ 0, d -1 in L 2 (0, T ; R) i.e.
T 0 (-t) p e -λ k,n t q l j,n (t)dt = δ k,j δ p,l , ∀k, j ∈ N * , ∀p, l ∈ 0, d -1 , which satisfies

q l k,n L 2 (0,T ) ≤ C T,ε,d e ελ k,n , ∀k ∈ N * , ∀l ∈ 0, d -1 . Proof. Let ε > 0 and d ≥ 2. Let τ ∈ 3 4 , 1 be such that 3d(1 -τ ) ≤ ε 2 , (4.5)
and γ τ the constant given by Corollary (1.2). The rest of the proof follows exactly the proof of Proposition 4.2 but we apply the abstract construction given in Theorem 3.2 instead of Theorem 1.3. Again, if λ k,n ∈ L τ,n ∪ U n , the estimate follows directly whereas, if λ k,n ∈ M τ,n the main contribution comes from the term

1 ρ λ dc γτ λ ≤ exp 3d(1 -τ )λ k,n + 6d λ k,n ≤ e ε 2 λ k,n +6d √ λ k,n ,
where we used (4.5).

We now turn to the proof of Theorem 1.4.

Proof (of Theorem 1.4). Let T > a 2 2 . Following the method presented in Section 1.3, for every n ∈ N * , it suffices to find a null control u n , uniformly bounded with respect to n, to the following 1D system

     ∂ t F n (t, x) + O n F n (t, x) = B d u n (t, x)1 (a,b) (x), (t, x) ∈ (0, T ) × (-1, 1), F n (t, ±1) = 0, t ∈ (0, T ), F n (0, x) = F 0 n (x) ∈ (L 2 (-1, 1)) d , x ∈ (-1, 1), (4.6)
where we have introduced the operator

O n := A n I d + K d .
The reduction of the null controllability of cascase systems like (4.6) to a moments problem as well as its formal resolution cand be found for instance in [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF]Section 5.1]. For an adaptation with a distributed control as considered in (4.6), we refer for instance to [2, Proof of Theorem 5.2].

Following these references, we consider the Hilbert basis of

L 2 (-1, 1; R) d of gen- eralized eigenvectors Φ r k,n (x) := g k,n (x)e r , ∀r ∈ 1, d , k, n ∈ N * ,
where e r is the r th vector of the canonical basis of R d i.e.

(O * n -λ k,n )Φ r k,n = Φ r-1 k,n , (4.7)
where Φ 0 k,n = 0. With this notation, a formal null control of (4.6) is given by

u n (t, x) = - k∈N * d r=1 (r -1)! F 0 n , e -O * n T Φ r k,n L 2 (-1,1) g k,n (x) g k,n 2 L 2 (a,b) q r-1 k,n (T -t).
From (4.7), we obtain

e -O * n t Φ r k,n = e -λ k,n t r-1 s=0 (-t) s s! Φ r-s k,n , which implies F 0 n , e -O * n T Φ r k,n L 2 (-1,1) ≤ C d e -λ k,n T F 0 n L 2 (-1,1) .
Then, null controllability in time T follows exactly as in the proof of Theorem 1.1.

Extension to boundary controls.

In this section we detail how to adapt our analysis to deal with boundary controls. This leads to Theorem 1.5. First, let us consider the scalar case i.e. d = 1.

Using a classical argument based on cut-off functions, the lack of null controllability in time T < 1 2 follows directly from Theorem 1.1. Now, let T > 1 2 and ε > 0 be such that T > 1 2 + 2ε. As previously, null controllability of (1.2) is equivalent to the uniform (with respect to the parameter n) controllability of the following family of problems

         ∂ t f n (t, x) + A n f n (t, x) = 0, (t, x) ∈ (0, T ) × (-1, 1), f n (t, -1) = 0, t ∈ (0, T ), f n (t, 1) = u n (t), t ∈ (0, T ), f n (0, x) = f 0 n (x), x ∈ (-1, 1). ( 4.8)
The moments problem equivalent to the null controllability of (4.8) now reads

g k,n (1)
T

0 e -λ k,n (T -t) u n (t)dt = -f 0 n , e -λ k,n T g k,n L 2 (-1,1) , ∀k ∈ N * . (4.9)
A formal solution of this countable family of equations is given by

u n (t) = - k∈N * e -λ k,n T f 0 n , g k,n L 2 (-1,1) g k,n (1) q k,n (T -t), (4.10)
where, for any n ∈ N * , (q k,n ) k∈N * is the biorthogonal family in L 2 (0, T ; R) to

t → e -λ k,n t ; k ∈ N *
given in Proposition 4.2. Thus, there exists C T,ε > 0 such that

q k,n L 2 (0,T ) ≤ C T,ε e ελ k,n , ∀k ∈ N * , ∀n ∈ N * . ( 4
ε > 0 such that |g k,n (1)| ≥ C ε e -λ k,n ( 1 2 +ε) , ∀k ∈ N * , ∀n ∈ N * .
Proof. Using a classical argument of extension of the spatial domain, the analysis of the minimal null control time done in [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF] implies the null controllability in any time T > 1 2 of the boundary control problem

         ∂ t f (t, x, y) + Df (t, x, y) = 0, (t, x, y) ∈ (0, T ) × Ω f (t, x, y) = 0, (t, x, y) ∈ (0, T ) × (∂Ω\Γ) f (t, x, y) = u(t, y), (t, x, y) ∈ (0, T ) × Γ f (0, x, y) = f 0 (x, y) ∈ L 2 (Ω), (x, y) ∈ Ω, (4.12)
where Γ = Γ -1 ∪ Γ 1 with Γ ±1 = {±1} × [0, 1]. Thus, this problem satisfies the quantitative Fattorini-Hautus (proved in [19, Proposition 2.3] and studied in more details in [START_REF] Khodja | Quantitative Fattorini-Hautus test and minimal null control time for parabolic problems[END_REF]) which is a necessary condition for null controllability in time T . In particular, for any ε > 0, there exists C ε > 0 such that every normalized eigenvector ϕ λ of D associated with the eigenvalue λ satisfies

∂ ν ϕ λ 2 L 2 (Γ) ≥ C ε λe -2λ( 1 2 +ε) .
Applying this inequality to (x, y) → g k,n (x)φ n (y) we obtain

|g k,n (-1)| 2 + |g k,n (1)| 2 ≥ C ε λ k,n e -2λ k,n ( 1 2 +ε) , ∀k ∈ N * , ∀n ∈ N * .
As the eigenfunctions g k,n are either odd or even (see Lemma 2.1) this ends the proof of Proposition 4.4.

Recall that, from Proposition 2.1,

λ k,n ≥ nπ(2k -1) ≥ 2k -1, ∀k ∈ N * , ∀n ∈ N * .
Then, using Proposition 4.4 and estimate (4.11) in (4.10) leads to the existence of C T,ε > 0 such that

u n L 2 (0,T ) ≤ C T,ε k∈N * e -λ k,n T e λ k,n ( 1 2 +ε) e ελ k,n f 0 n , g k,n L 2 (-1,1) ≤ C T,ε k∈N * e -(2k-1)(T -1 2 -2ε) f 0 n L 2 (-1,1)
which ends the proof of Theorem 1.5 in the case d = 1.

The adaptation to the minimal null control time of the cascade system (1.23) follows exactly the lines of Section 4.2.

Appendix.

Proof of Lemma 2.3.

Recall that the functions (G k ) k≥1 are normalized in L 2 (R) and satisfy

G k (x) = e -x 2 /2 H k (x) π 1/4 2 k-1 2 (k -1)! , (5.1)
where H k is the (k -1) th Hermite polynomial defined by

H k (x) = (-1) k-1 e x 2 d k-1 dx k-1 e -x 2
, ∀k ≥ 1.

These polynomials are orthogonal for the following scalar product Then, we use (5.1) to get:

(f, g) = R f (
G k (x) = 2x
e -x 2 /2 H k-1 (x)

π 1/4 2 k-1 2 (k -1)! - 2(k -2) π 1/4 2 k-1 2 (k -1)! e -x 2 /2 H k-2 (x) = 2 k -1 xG k-1 (x) - √ k -2 √ k -1 G k-2 (x).
G k+1 (x) = 2 k xG k (x) - k -1 k G k-1 (x). (5.3)
Thus,

G k+1 (x) ≥ 2 k x - k -1 k G k (x).
(5.4) Since x ≥ √ nπ, we have

2 k x - k -1 k ≥ 2 k √ nπ -1 ≥ 2 k √ 2k -1 ≥ 1, since k ≤ πn 2 .
As G k (x) > 0, this gives (5.2). According to inequality (5.2), Lemma 2.3 is proved, if for k 0 := τ n π 2 , the following inequality holds As τ ∈ (0, 1), let α > 0 be such that cosh(α) = 1 τ . Let x 0 := √ µ k0 cosh(α).

Observe that, for any n ∈ N * , √ nπ ≥ x 0 ≥ √ µ k0 .

Then, the estimate given in [29, formula (8.22.13), p. 201] reads G k0 (x 0 ) = exp ( k0 2 -1 4 )(2α -sinh(2α) √ π2 3/4 (k 0 -1) 1/4 sinh(α)

1 + O τ 1 n ,
where O τ 1 n only depends on τ and n. Recall that α > 0, so using that 2αsinh(2α) < 0, we find, for n ≥ n 0 with n 0 large enough,

G k0 (x 0 ) ≤ C 1 exp(-C 2 n),
where C 1 et C 2 only depend on τ .

Remark 5.1. Lemma 2.3 is the only step in the whole spectral analysis that requires the introduction of the parameter τ . This is what introduces a discrepancy between the analysis of low and high frequencies and led us to the generalization of biorthogonal families stated in Theorem 1.3. In the case τ = 1, instead of Lemma 2.3, we only obtain a polynomial decay (with respect to n) of G k (x) in the considered range for k and x. This decay would not be sufficient to repeat the proof of the uniform gap property fo low frequencies (see for instance (2.19)).

Proof of Lemma 2.2.

We shall in fact prove the following slightly more general property: for any δ ∈ (0, 1), there exists an integer n 0 (δ) ∈ N * such that Recall that µ k = 2k -1. Let δ ∈ (0, 1) and ε ∈ (0, π 2 ) to be defined later, but depending only on δ. Let k ≤ π 2 n, we have (5.7)

√ nπ ≥ √ µ k ≥ √ µ k cos(ε). Since G 2 k is even,
Let us focus on the term that does not involve r ε,k . Let

v k : φ → k - 1 2
(sin(2φ) -2φ) + 3π 2 .

We have

4 π π 2 ε sin 2 v k (φ) 2 dφ = 2 π 1 - 2 π π 2 ε cos (v k (φ)) dφ = 1 -ε 2 π - 2 π π 2 ε cos (v k (φ)) dφ.
To estimate this term, we will fix ε sufficiently small (depending on δ) and prove that the remaining integral goes to 0 when k goes to +∞. Indeed, integrating by parts, we obtain As K depends only on δ, we deduce that there exists n 0 depending only on δ such that

4 π π 2 ε sin 2 (2v k (φ)) dφ = 1 -ε 2 π - 2 π π 2 ε cos (v k (φ)) v k (φ) v k (φ) dφ = 1 -ε 2 π - 2 π sin(v k (φ) v k (φ)
Ωn G 2 k (x)dx ≥ 1 -δ, ∀k ≤ π 2 n, ∀n ≥ n 0 ,
which is exactly the generalized statement of Lemma 2.2 announced at the beginning of Section 5.2.

Figure 1 . 1 :

 11 Figure 1.1: The geometrical setting

Figure 1 . 2 :

 12 Figure 1.2: Gap divided by nπ : k → λ k+1,n -λ k,n nπ .

  .22) where F is now a vector of R d and I d is the identity matrix of size d. The coupling matrix K d of size d × d and the control vector B d ∈ R d are defined as follows

Following [ 2 ,

 2 Lemma 2.2 and proof of Lemma 2.3], if we set W = w w √ λ

  /2) > 1, then in particular γ < ρ λ and we deduce that, we necessarily have c γ λ = 0, so that (3.7) is still valid and actually reduces to D 3 (x) ≥ 1.• The last terms D 2 (x) and D 4 (x) are dealt with in the exact same way. Let us detail the treatment of D 4 . From Lemma 3.1 it comes that log D 4 (x)

Remark 3 . 4 .

 34 As detailed for Proposition 3.1, the same estimate as (3.15) holds without the log(λ) term in the exponential if one replaces assumption (1.20) by (3.2).

2 )

 2 x)g(x)e -x 2 dx.Those functions are drawn in Figure5.1.Let us start by showing0 < G k (x) ≤ G k+1 (x), ∀x ≥ √ nπ, ∀k ∈ 1,One can observe this inequality in Figure5.1.From the definition of Hermite polynomials, one can derive easily the classical recurrence relation (see for instance [29, Section 5.5])H k (x) = 2xH k-1 (x) -2(k -2)H k-2 (x), ∀k ≥ 2.

6 Figure 5 . 1 : 2 π 1 / 4 √ 2 ,

 6512142 Figure 5.1: Graph of some eigenfunctions of the harmonic oscillator.

  G k0 ( √ nπ) ≤ C 1 e -C2n .(5.5)Let us show this inequality. First, for anyk ∈ N * , G k is decreasing in [ √ µ k , +∞). Indeed, when x ≥ √ µ k , we have proved that G k (x) > 0. Moreover, for a such x, G k (x) = (x 2 -µ k )G k (x) ≥ 0. Thus, the function G k is convex in [ √ µ k , +∞[ andtends to zero at infinity. It follows that G k is decreasing in this interval.It suffices now to show that G k0 (x 0 ) ≤ C 1 e -C2n for a x 0 such that √ nπ ≥ x 0 ≥ √ µ k0 . Recall that µ k = 2k -1.

Ωn G 2 k

 2 (x)dx ≥ 1 -δ, ∀k ≤ π 2 n, ∀n ≥ n 0 .

≥ 2 √ µ k cos(ε) 0 G 2 k 2 k√

 2022 (x)dx. Let us make the change of variable x = √ µ k cos(φ) with φ ∈ [ε, π µ k cos(φ) √ µ k sin(φ)dφ. For any φ ∈ (ε, π 2 ), the following formula for G k ( √ µ k cos(φ)) holds (see [29, Theorem 8.22.9, p. 201]) ε,k , (5.6) where r ε,k only depends on ε and k and, for ε fixed, r ε,k = O

π 2 εNow let us fix ε > 0 such that ε 2 π = δ 3 . 1 k 3 ,

 2313 sin(v k (φ))v k (φ)(v k (φ)) There exists K ∈ N * such that 2 that K depends only on ε and thus only on δ. Finally, let us deal with the remaining terms in (5.7). As ε is fixed and r ε,k = O k→+∞ , it comes that, there exists K δ ∈ N * depending only on δ such that ∀k ≥ K δ .Using this estimate and (5.8) in (5.7), we obtain that there exists K ∈ N * depending only on δ such thatΩn G 2 k (x)dx ≥ 1 -δ, ∀n, k ∈ N * such that π 2 n ≥ k ≥ K. (5.9) Considering the finite number of k ∈ 1, K -1 , as G k L 2 (R) = 1, it comes that Ωn G 2 k (x)dx → n→+∞ 1.

  .11) The only thing left to prove unifom null controllability of (4.8) in time T is to give a suitable lower bound on |g k,n (1)|. This lower bound is given in the following proposition.

Proposition 4.4

. For any ε > 0, there exists C

Proof. The proof relies on the Cauchy formula and Proposition 3.1. Let Γ λ be the circle centered at λ and of radius r λ := min(γ,ρ λ )


Jean-Bernard Chatelain
email: jean-bernard.chatelain@univ-paris1.fr

Kirsten Ralf
email: kirsten.ralf@esce.fr.

Ramsey Optimal Policy versus Multiple Equilibria with Fiscal and Monetary Interactions

Keywords: JEL classification numbers: E63, C61, C62, E43, E44, E47, E52, E58 Frictionless endowment economy, Fiscal theory of the Price Level, Ramsey optimal policy, Interest Rate Rule, Fiscal Rule

. Under this formulation there exists a unique Ramsey equilibrium, with an interest rate peg and a passive fiscal policy.

Introduction

Monetary and fiscal interactions are often presented using [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF] model of frictionless endowment economies. This model includes the intertemporal budget constraint of the government and a Fisher relation with a constant real interest rate equal to the representative household's discount factor. In [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF], for monetary policy, the interest rate responds in proportion to inflation. For fiscal policy, a lump-sum tax responds in proportion to the stock of real public debt.

Assuming inflation, interest rate and lump-sum tax are forward-looking variables for seeking equilibria with [START_REF] Blanchard | The solution of linear difference models under rational expectations[END_REF] determinacy condition, [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF] obtains two equilibria for monetary and fiscal interactions. These two equilibria are defined by ranges of values of the parameters of ad hoc policy rules. In the first equilibrium, the interest rate rule parameter destabilizes inflation ("active monetary policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF]) and the fiscal rule parameter stabilizes public debt ("passive fiscal policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF]). In the second equilibrium related to the fiscal theory of the price level, the fiscal rule parameter destabilizes public debt ("active policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF]) and the interest rate rule parameter stabilizes inflation ("passive monetary policy" according to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF]). A peg of the interest rate with a lack of response of the interest rate to inflation is a particular case of a "passive monetary policy" equilibrium.

This note demonstrates that Ramsey optimal policy is a third equilibrium with an optimal interest rate peg and "passive" fiscal policy.

Section two presents the policy transmission mechanism, the two equilibria with ad hoc policy rules and the Ramsey optimal policy equilibrium. Section three concludes. Consider an infinitely-lived representative consumer who receives a constant endowment of goods each period in the amount y and derives utility only from consumption c t . The government purchases a constant quantity of goods g > 0 from the consumer each period. We impose equilibrium in the goods market, so that consumption: c t = y -g.

The consumer makes a consumption-saving decision that produces the Fisher relation where the real rate is here equal to a constant discount rate:

E t 1 π t+1 = 1 β 1 R t (1)
where R t is both the gross one-period nominal interest rate on nominal bonds bought at t and pay off in t + 1 and the monetary policy instrument, and π t+1 = P t+1 /P t is the gross rate of inflation between t and t + 1, with P t the aggregate price level. In the steady state, P t = P t+1 ⇒ π * = 1 where π * is the inflation target. Then, R * = π * /β = 1/β, is the nominal interest rate consistent with the inflation target according to the Fisher relation. The equilibrium real interest rate is constant at r = (1/β) -1 where 0 < β < 1 is the consumer's discount factor. The Fisher relation in deviation of steady state values is:

E t 1 π t+1 - 1 π * = 1 β 1 R t - 1 R * ⇒ E t 1 π t+1 -1 = 1 β 1 R t -β . ( 2
)
It is linearized around the steady state equilibrium:

E t π t+1 -π * = β (R t -R * ) ⇒ E t π t+1 -1 = β R t - 1 β .
Fiscal policy levies lump-sum taxes of τ t and sets purchases to be constant, g > 0 with primary surplus s t = τ t -g. Government issues one-period nominal bonds, B t , that satisfy the flow constraint, where P t is the aggregate price level and real debt is defined as b t = B t /P t .

b t = B t P t = -(τ t -g) + R t-1 P t-1 P t B t-1 P t-1 = -(τ t -g) + R t-1 π t b t-1 .
Using the Fisher relation, we substitute the constant real interest rate in the government intertemporal budget constraint:

b t = -(τ t -g) + 1 β b t-1 .
The dynamics of real debt does not depend on inflation or on the nominal rate R t . The steady state level of real government debt has an exogenous value: b t+1 = b t = b * . To be consistent with the government intertemporal budget constraint, the steady state level of tax revenue τ * is equal to the steady state interest expense:

τ * -g = 1 β -1 b * .
The government intertemporal budget constraint written in deviation of steady state is

b t -b * = -(s t -s * ) + 1 β (b t-1 -b * ).
The linearized dynamics are (in deviation from steady state):

E t π t+1 -π * E t b t+1 -b * = A π = 0 A πb = 0 A bπ = 0 A b = 1 β =A π t -π * b t -b * + B πR = β B πs = 0 B bR = 0 B πs = -1 =B R t -R * s t+1 -s * .
The log-linearized dynamics of inflation is:

E t π t+1 -π * π * = β R * π * R t -R * R * = R t -R * R * .
The log-linearized dynamics of real debt is:

E t b t+1 -b * b * = 1 β b t -b * b * - s * b * s t+1 -s * s * with τ * -g b * = 1 β -1.
Log-linearized dynamics are:

Etπ t+1 -π * π * Etb t+1 -b * b * = A π = 0 A πb = 0 A bπ = 0 A b = 1 β =A πt-π * π * bt-b * b * + 1 0 0 -1 β -1 =B log Rt-R * R * s t+1 -s * s *

Ad Hoc Policy Rules

The fiscal authority adjusts lump-sum tax in response to the level of real government debt. Monetary policy follows an interest rate rule that responds to inflation:

s t -s * = G b (b t-1 -b * ) + ε s t and R t -R * = F π (π t -π * ) + ε R t Shocks ε R t , ε s
t are assumed to be independently and identically distributed, with mean zero and a non-zero variance-covariance matrix. The linearized dynamics are:

E t 1 π t+1 -1 π * E t-1 b t+1 -b * = 1 β F π 0 0 1 β -G b π t -π * b t -b * + β 0 0 -1 ε R t ε s t with b 0 given.
Public debt is the only predetermined variable. Blanchard and Kahn's (1980) determinacy condition implies that one eigenvalue should be inside the unit circle and one should be outside the unit circle. Either the inflation eigenvalue is outside the unit circle |βF π | > 1 and the public debt eigenvalue 1 β -G b < 1 is inside the unit circle (first equilibrium) or it is the reverse (second equilibrium):

|βF π | < 1 and 1 β -G b > 1 (Leeper (1991)).

Ramsey Optimal Policy under Quasi-Commitment

In a monetary policy regime indexed by j, a policy maker may re-optimize on each future period with exogenous probability 1 -q strictly below one. Following [START_REF] Schaumburg | An investigation of the gains from commitment in monetary policy[END_REF], we assume that the mandate to minimize the loss function is delegated to a sequence of policy makers with a commitment of random duration. The degree of credibility is modelled as if it is a change of policy-maker with a given probability of reneging commitment and re-optimizing optimal plans. The length of their tenure or "regime" depends on a sequence of exogenous independently and identically distributed Bernoulli signals {η t } t≥0 with E t [η t ] t≥0 = 1 -q, with 0 < q ≤ 1. If η t = 1, a new policy maker takes office at the beginning of time t. Otherwise, the incumbent stays on. A higher probability q can be interpreted as a higher credibility. A policy maker with little credibility does not give a large weight on future welfare losses. The policy maker j solves the following problem for regime j, omitting subscript j, before policy maker k starts:

V j 0 = -E 0 t=+∞ t=0 (βq) t 1 2 Q π (π t -π * ) 2 + Q b (b t -b * ) 2 + µ R (R t -R * ) 2 + µ s (s t -s * ) +β (1 -q) V k t R t -R * = βq (E t π t+1 -π * ) + β (1 -q) E t π k t+1 -π * b t -b * = βq(E t b t+1 -b * ) + β (1 -q) E t b k t+1 -b * + βq(s t -s * ) + β (1 -q) (s k t -s * ).
Inflation and public debt expectations are an average between two terms. The first term, with weight q is the inflation and public debt, respectively, that would prevail under the current regime upon which there is commitment. The second term with weight 1 -q is the inflation and public debt, respectively, that would be implemented under the alternative regime by policy maker k.

This optimal program is a discounted linear quadratic regulator (LQR) with a "credibility adjusted" discount factor βq. The log-linear version of the dynamic system can also be used instead of the linear version. Preferences of the policy maker are firstly given by positive weights for the two policy targets

Q π ≥ 0, Q π ≥ 0 (Q = diag(Q π , Q b )).
Secondly, there are at least two non-zero policy maker's preferences for interest rate smoothing and primary surplus and tax smoothing, with strictly positive weights for these two policy instruments in the loss function: µ R > 0, µ s > 0 (R = diag(µ R , µ s )). This ensures the strict concavity of the LQR program which implies the determinacy (uniqueness) of the solution of this Stackelberg dynamic game, if the system of the transmission mechanism is controllable.

All the results are valid for nearly negligible cost of changing the policy instruments: a minimal interest rate smoothing parameter (µ R ≥ 10 -7 ), a minimal tax smoothing parameter (µ s ≥ 10 -7 ) and a minimal credibility (a non zero probability of not reneging commitment next period: 10 -7 ≤ q ≤ 1).

If µ R = 0 or µ s = 0 or if the costs of changing the policy instruments are not strictly convex or if q = 0, the results are no longer valid. If q = 0, the policy maker knows that he is replaced next period. He does static optimization ((βq) 0 = 1) of his current period quadratic loss function subject to a static transmission mechanism where he considers the expectations terms related to the next period policy maker as exogenous intercepts (Chatelain and Ralf (2019b)).

Proposition 1 For the transmission mechanism of the Fisher relation with constant real rate and the government intertemporal budget constraint, Ramsey optimal policy has a unique equilibrium. The interest rate is pegged at its long run value R t = R * . A feedback Taylor rule is not optimal and the Taylor principle should not be satisfied: F * = 0 < β. This is a "passive monetary policy". The optimal auto-correlation of monetary policy shocks is zero: ρ * ε R = 0. The optimal variance of monetary policy shocks is zero σ 2 ε R = 0. Inflation jumps to its steady state value instantaneously following monetary policy shocks π * 0 = 0, as in a degenerate rational expectations model without predetermined variables. Hence, the price level is constant. Ramsey optimal fiscal rule has a negative feedback parameter ("passive fiscal policy") with ensures the local stability of public debt dynamics. There are two stable eigenvalues giving the optimal persistence of inflation (λ * π = 0) and the optimal persistence of public debt (0 < λ * b < 1).

Proof. Ramsey optimal policy amounts to find the solution of a linear quadratic regulator (Chatelain and Ralf (2019a)). The autocorrelation parameters of policy maker's shocks are optimally set to zero, else they increase the volatility of inflation and of public debt in the policy maker's loss function. The optimal expected value of the loss function is:

L * = - 1 2 π * 0 b 0 P π * 0 b 0 T with P = P π P πb P πb P b , ( 3
)
π * 0 = π * 0 -π * and b 0 = b 0 -b * . ( 4
)
where P is a positive symmetric square matrix of dimension two which is the solution of the discrete algebraic Riccati equation (DARE) of the LQR. The optimal initial anchor of inflation π * 0 on public debt b 0 is given by:

∂L * ∂π t t=0 = P π π * 0 + P πb b 0 = 0 ⇒ π * 0 = P πb P π b 0 if P π = 0. (5)
This initial transversality (or natural boundary) condition eliminates the indeterminacy of initial inflation π 0 put forward for the ad hoc policy rules solution. Because the system is decoupled (A πb = A bπ = 0) and because the weight on the product π t b t is zero in the loss function Q πb = 0, this implies a zero weight P πb = 0 in the optimal value of the loss function. Therefore, P π ≥ 0 and P b ≥ 0 are solutions of scalar discrete algebraic Riccati equations (DARE). Because the system of the policy transmission mechanism is decoupled, the optimal program is identical if we consolidate the central bank and the treasury as a single policy maker or if we consider that they are distinct policy makers. For inflation, this DARE equation is:

P π = Q π + βA ′ π P π A π -β ′ A ′ π P π B πR (µ R +βB ′ πR P π B πR ) -1 βB ′ πR P π A π = Q π .
Because A πb = A bπ = A π = 0, this implies that the optimal loss function parameter P π is equal to the inflation weight in the loss function: P π = Q π . Therefore, if the policy maker has a non-zero weight on inflation volatility in his loss function Q π > 0, optimal initial inflation is zero because P πb = 0: π * 0 = 0. Because inflation dynamics are decoupled from the dynamics of predetermined public debt, inflation behaves exactly as in a degenerate rational equilibrium model where there is no predetermined variable. At any date t, if ever there is a monetary policy shock ε R t , inflation instantaneously jumps back to equilibrium π * t = 0. There are no transitory dynamics. The volatility of inflation is zero.

Because the open-loop system is already at the zero lower bound of inflation persistence (A π = 0) and because there is a non-zero cost of interest rate volatility (µ R > 0) in the loss function, it is optimal to set an interest rate peg R t = R * . Therefore, the quadratic term of interest rate volatility µ R (R t -R * ) 2 is minimized at its zero lower bound zero in the loss function. This implies a Taylor rule parameter equal to zero: F * = 0 < β, a zero auto-correlation of monetary policy shocks ρ ε R = 0, and a zero variance of monetary policy shocks σ 2 ε R = 0 initiated by the policy maker. If there is a zero cost of inflation in the loss function (Q π = 0), the policy maker does not care about inflation. There is an indeterminacy for the optimal choice of the initial value of inflation π * 0 . However, the implied volatility originated by this indeterminacy does not matter in the policy maker's loss function.

For public debt, the solution is a scalar case of Ramsey optimal policy under quasicommitment (we use the linear dynamics instead of the log-linear dynamics).

E t b t+1 -b * = 1 √ βq (b t -b * ) -βq(s t -s * ) with A b = 1 βq or βqA = 1 and B bs = -1.
Optimal public debt persistence (or auto-correlation or closed-loop eigenvalue) is the stable root of the characteristic polynomial of the Hamiltonian system:

0 = λ 2 -Sλ + 1 βq with λ * λ 2 = 1 βq with S = A + 1 Aβq + B 2 A Q b µ s = 1 + 1 βq + βq Q b µ s 0 < λ * b (µ s , βq) = 1 2 S -S 2 - 4 βq < 1.
The optimal public debt persistence or autocorrelation λ * b increases with the cost µ s of changing the policy instrument (lump sum taxes). Its boundaries are given by limits when the cost of changing policy instrument µ s tends to zero and when tends µ s to infinity:

lim µs→0 λ * b (µ s , βq) = lim µ b →0 1 2 βq µ s - βq µ s = 0 and lim µs→+∞ λ * b (µ s , βq) = 1 βqA = 1.
The fiscal rule parameter G b remains in the range of values so that the persistence of public debt is strictly positive and strictly below one:

1 βq -1 < G * b = λ * b -A B = 1 βq -λ * b < 1 βq .
The optimal loss function parameter P b is:

P b = 1 A Q b qβ 1 -λ * b = Q b 1 -λ * b .
It is also the solution of a scalar algebraic Riccati quadratic equation. For a given initial value of predetermined public debt b 0 and for preferences Q b ≥ 0 which can be equal to zero, the optimal expected value of the policy maker loss function is:

If Q π > 0 : L * = - 1 2 0 b 0 Q π 0 0 Q b 1-λ * b 0 b 0 T = - 1 2 Q b 1 -λ * b (b 0 -b * ) 2 . If Q π = 0 : L * = - 1 2 π * 0 b 0 0 0 0 Q b 1-λ * b π * 0 b 0 T = - 1 2 Q b 1 -λ * b (b 0 -b * ) 2 .
A rational policy maker with quadratic preferences including a convex cost of changing policy instruments (interest rate smoothing µ R > 0 and tax smoothing µ s > 0) taking into account private sector expectations with a minimal credibility (q > 0) eliminates the indeterminacy of initial inflation π 0 if the policy maker's preferences includes a non-zero weight (Q π > 0) on inflation volatility in his loss function. Else, a rational policy maker neglects the indeterminacy of initial inflation because inflation volatility has zero weight (Q π = 0) in his loss function. A SCILAB code is available from the authors to solve Ramsey optimal policy with numerical values.

Conclusion

For the model of frictionless endowment economies, Ramsey optimal policy is an interest rate peg and a "passive" fiscal policy. It is a third equilibrium with respect to [START_REF] Leeper | Equilibria under 'active'and 'passive'monetary and fiscal policies[END_REF] two equilibria with simple feedback rules.