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

J Vancostenoble

tex-file : martinez-vancostenoble-cost-singularity-15aout2017 THE COST OF BOUNDARY CONTROLLABILITY FOR A PARABOLIC EQUATION WITH INVERSE SQUARE POTENTIAL

The goal of this paper is to analyze the cost of boundary null controllability for the 1 -D linear heat equation with the so-called inverse square potential:

, it is known that such problems are well-posed for any µ ≤ 1/4 (the constant appearing in the Hardy inequality) whereas instantaneous blow-up may occur when µ > 1/4.

 (via Carleman estimates) that the equation can be controlled (in any time T > 0) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at x = 1. The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two paramaters T > 0 and µ ∈ (-∞, 1/4]. Our proofs are based on the moment method and very recent results on biorthogonal sequences. P. MARTINEZ AND J. VANCOSTENOBLE V.1. Bessel's equation and Bessel's functions of order ν 21 V.2. Fundamental solutions of Bessel's equation when ν / ∈ N 21 V.3. Fundamental solutions of Bessel's equation when ν = n ∈ N 21 V.4. Zeros of Bessel functions of order ν of the first kind 22 V.5. Orthogonality property 23 References 23

 [START_REF] Biccari | Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function[END_REF]. Existence of a suitable biorthogonal family III.6. Upper bounds of the cost: proof of (I. 4) and (I. 6) III.7. Lower bound of the cost when µ ∈ [0, 1 4 ]: proof of (I. 5) III.8. Lower bound of the cost when µ ≤ 0: proof of (I.

I. Introduction and main results

I.1. Description of the problem. In this paper, we are interested in the linear 1 -D heat equation with an inverse square potential (that arises for example in the context of combustion theory and quantum mechanics):

(I. 1)            u t -u xx - µ x 2 u = 0
x ∈ (0, 1), t ∈ (0, T ), u(0, t) = 0 t ∈ (0, T ), u(1, t) = H(t) t ∈ (0, T ), u(x, 0) = u 0 (x)

x ∈ (0, 1), where u 0 ∈ L 2 (0, 1), T > 0 and µ is a real parameter. Here H represents some control term that aims to steer the solution to zero at time T . Our goal is not only to establish the existence of such control (which could be easily deduced from known results, see later) but also to provide sharp estimates of the cost of such control.

Since the works by Baras and Goldstein [START_REF] Baras | Remarks on the inverse square potential in quantum mechanics[END_REF][START_REF] Baras | The heat equation with a singular potential[END_REF], it is known that existence/nonexistence of positive solutions is determined by the value of µ with respect to the constant 1/4 appearing in the Hardy inequality [START_REF] Hardy | Inequalities[END_REF][START_REF] Opic | Hardy-type Inequalities[END_REF]:

(I. 2)
∀z ∈ H 1 0 (0, 1), 1 4

1 0 z 2 |x| 2 dx ≤ 1 0 |z x | 2 dx.
When µ < 1/4, the operator z → -z xx -µx -2 z generates a coercive quadratic form in H 1 0 (0, 1). This allows showing the well-posedness in the classical variational setting of the linear heat equation with smooth coefficients. For the critical value µ = 1/4, the space H 1 0 (0, 1) has to be slightly enlarged as shown in [START_REF] Vázquez | The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential[END_REF] but a similar result of well-posedness occurs (see section II.2 for details). Finally, when µ > 1/4, the problem is ill-posed (due to possible instantaneous blow-up) as proved in [START_REF] Baras | Remarks on the inverse square potential in quantum mechanics[END_REF].

For these reasons, we concentrate on the two first cases and we assume throughout this paper that µ satisfies µ ≤ 1/4.

Recently, the null controllability properties of (I. 1) began to be studied. For any µ ≤ 1/4, it has been proved in [START_REF] Vancostenoble | Null controllability for the heat equation with singular inversesquare potentials[END_REF] that such equations can be controlled (in any time T > 0) by a locally distributed control: ∀µ ≤ 1/4, ∀u 0 ∈ L 2 (0, 1), ∀T > 0, ∀0 ≤ a < b ≤ 1, there exists h ∈ L 2 ((0, 1) × (0, T )) such that the solution of (I. 3)

           u t -u xx - µ x 2 u = h(x, t)χ (a,b) (x)
x ∈ (0, 1), t ∈ (0, T ), u(0, t) = 0 t ∈ (0, T ), u(1, t) = 0 t ∈ (0, T ), u(x, 0) = u 0 (x)

x ∈ (0, 1), satisfies u(•, T ) ≡ 0. The proof in [START_REF] Vancostenoble | Null controllability for the heat equation with singular inversesquare potentials[END_REF] is based on Carleman estimates. It also concerns the case of the N -dimensional equation with some restricting geometric condition on the region of the control, condition that has been later erased in [START_REF] Ervedoza | Control and stabilization properties for a singular heat equation with an inversesquare potential[END_REF]. After those first results, several other works followed extending them in various situations. See for instance [START_REF] Vancostenoble | Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems[END_REF][START_REF] Vancostenoble | Lipschitz stability in inverse source problems for singular parabolic equations[END_REF][START_REF] Cazacu | Controllability of the heat equation with an inverse-square potential localized on the boundary[END_REF][START_REF] Biccari | Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function[END_REF][START_REF] Fotouhi | Null controllability of degenerate/singular parabolic equations[END_REF][START_REF] Hajjaj | Carleman estimates and null controllability of degenerate/singular parabolic systems[END_REF].

Here we are insterested in the study of null controllability using a boundary control acting at x = 1 (see problem (I. 1)). More precisely, our aim is to provide sharp estimates of the cost of controllability in that case, analyzing its dependence with respect to the two parameters T > 0 and µ ∈ (-∞, 1 /4].

Observe that the existence of a boundary control for (I. 1) could be deduced from [START_REF] Vancostenoble | Null controllability for the heat equation with singular inversesquare potentials[END_REF], using for example the standard argument that consists in extending the domain (0, 1) into (0, 1 + η), applying the result of null controllability for (I. 3) in (0, 1 + η) with a distributed control localized in (1, 1 + η) and concluding by taking the trace at x = 1 of such a control function. However this method based on Carleman estimates would not provide optimal estimates of the cost of the control, in particular when µ → -∞ (see Remark 3.5 in [START_REF] Vancostenoble | Null controllability for the heat equation with singular inversesquare potentials[END_REF]).

For this reason, we turn here to methods based on decomposition in series and moment problems. These methods have been developped by Fattorini-Russell [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF], and have been successfully applied/adapted to obtain sharp results in quite simple geometric situations, we refer for example the reader to [START_REF] Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF][START_REF] Benabdallah | Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the N -dimensional boundary null controllability in cylindrical domains[END_REF][START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF][START_REF] Coron | Singular optimal control: A linear 1-D parabolic-hyperbolic example[END_REF][START_REF] Glass | A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit[END_REF][START_REF] Gueye | Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations[END_REF][START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF][START_REF] Lissy | On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension[END_REF][START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transportdiffusion equation[END_REF][START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF][START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF][START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF].

Here we follow the classical strategy:

• first of all, we transform the controllability question into a moment problem;

• to solve this moment problem and get some upper estimate of the cost of the control, we construct some suitable biorthogonal family (with best possible upper bounds of the norm of its elements); • to prove the sharpness of our estimate, we provide some lower bound of the cost of the control using lower bounds that are satisfied by any biorthogonal family.

Hence the tools that are required to solve our problem are :

• the construction of some suitable biorthogonal family (with best possible upper bounds of the norm of its elements); • lower bounds of the norm of the elements of any biorthogonal family.

As seen in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF], the obtention of explicit and precise (upper and lower) estimates for such biorthogonal families is closely related to gap conditions on the eigenvalues, namely ∀n,

γ min ≤ λ n+1 -λ n ≤ γ max .
(Roughly speaking, the gap γ min gives the upper estimates whereas the gap γ max gives the lower estimates). Then here

• we solve the eigenvalue problem, and we express the eigenvalues and eigenfunctions using Bessel functions and their zeros; • when µ ∈ [0, 1/4], the gap between the square root of successive eigenvalues satisfies some upper and lower bounds which are uniform with respect to the parameter µ; this enables us to use the results of [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF]; • but when µ ≤ 0, the eigenvalues do not satisfy a good uniform gap condition from above; we use here some new tools developped in [START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF] in order to treat cases where the eigenvalues do not satify a good uniform gap condition but satisfy some better asymptotic gap condition; these new results have already been applied in the context of the degenerate heat equation with a strong degeneracy in [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF], and in the present case they help us to provide a suitable lower bound of the cost.

I.2. Main results and comments.

Let us define the notion of cost of controllability. For any T > 0, µ ≤ 1/4 and u 0 ∈ L 2 (0, 1), we introduce the set of admissible controls:

U ad (µ, T, u 0 ) := {H ∈ H 1 (0, T ) | u (H) (T ) = 0},
where u (H) denotes the solution of (I. 1). Then we consider the controllability cost for any u 0 ∈ L 2 (0, 1)

C H 1 (µ, T, u 0 ) := inf H∈U ad (µ,T,u0) H H 1 (0,T )
which is the minimal value to drive u 0 to 0. Finally, we define the global notion of controllability cost:

C H 1 bd-ctr (µ, T ) := sup u0 L 2 (0,1) C H 1 (µ, T, u 0 ).
Then we prove the following results

Theorem I.1. Given µ ∈ [0, 1 4 ], T > 0, and u 0 ∈ L 2 (0, 1), there exists H ∈ H 1 (0, T ) such that the solution of (I. 1) satisfies u(•, T ) ≡ 0. Moreover, we have the following estimates of the controllability cost: there exists 0 < c < C both independent of µ ∈ [0, 1/4] and of T > 0 such that

(I. 4) C ctr-bd (µ, T ) ≤ Ce C/T e -(1+ √ 1 4 -µ) 2 T /C 1 + 1 4 -µ , and
(I. 5) C ctr-bd (µ, T ) ≥ ce c/T e -(1+ √ 1 4 -µ) 2 T /c .
Theorem I.2. Given µ ≤ 0, T > 0, and u 0 ∈ L 2 (0, 1), there exists H ∈ H 1 (0, T ) such that the solution of (I. 1) satisfies u(•, T ) ≡ 0. Moreover, we have the following estimates of the controllability cost: there exists 0 < c < C both independent of µ ≤ 0 and of T > 0 such that

(I. 6) C ctr-bd (µ, T ) ≤ Ce C/T e -(1+ √ 1 4 -µ) 2 T /C 1 + 1 4 -µ , and
(I. 7) C ctr-bd (µ, T ) ≥ ce c/T e -(1+ √ 1 4 -µ) 2 T /c e - √ 1 4 -µ 4/3 (ln √ 1 4 -µ+ln 1 T )/c .
Remark I.1. Several observations can be made from Theorems I.1 and I.2:

• we deduce from (I. 4) that the null controllability cost is uniformly bounded when µ ∈ [0, 1 4 ], in particular it does not blow-up as µ → 1/4; this is natural and expected since it has been proved in [START_REF] Vancostenoble | Null controllability for the heat equation with singular inversesquare potentials[END_REF] (in the case of a distributed control) that null controllability holds true for any µ ≤ 1/4; however, the proof of (I. 4) holds on the fact that 1 4 -µ ≥ 0, and the final estimate (I. 4) puts this in evidence; it also implies the following simpler estimate C ctr-bd (µ, T ) ≤ 2Ce C/T e -T /C , but where the condition 1 4 -µ ≥ 0 is hidden; • (I. 5) allows to measure how good is the qualitative behavior given by the upper bound (I. 4), once again putting in evidence that the condition 1 4µ ≥ 0 is not to be forgotten; the proofs of (I. 4) and of (I. 5) give explicit values of all the coefficients, hence in particular of the coefficients of 1/T in the exponential factors, but in this paper we are mainly interested in the qualitative behavior with respect to µ and T ; once again, (I. 5) implies the following simpler estimate C ctr-bd (µ, T ) ≥ ce c/T e -2T /c , but where the condition 1 4 -µ ≥ 0 is hidden;

• (I. 6) gives an upper bound interesting when T → 0, T → ∞, and µ → -∞; it implies the more compact form

C ctr-bd (µ, T ) ≤ Ce C/T e -(1+|µ|)T /C 1 + |µ| ,
but we stated (I. 6) to put it in parallel with (I. 4); note that (I. 6) implies that C ctr-bd (µ, T ) → 0 as µ → -∞: this comes from the fact the parameter here has the good sign (it makes the energy decreasing); this could not have been deduced from the result obtained by Carleman estimates since those estimates does not allow us to take into account the "good" or "bad" sign of potential terms (the constant appearing in Carleman estimates would grow up as µ → -∞, see Remark 3.5 in [START_REF] Vancostenoble | Null controllability for the heat equation with singular inversesquare potentials[END_REF]); • (I. 7) allows to measure how good is the qualitative behavior given by the upper bound (I. 6), the main difference coming from the last exponential factor; this allows us to precise the behavior of C ctr-bd (µ, T ) → 0 as µ → -∞.

µ := 1 + √ 1 -4µ 2 .
Observe that p(0) = 0, p(1) = 1 and

p"(x) + µ x 2 p(x) = 0.
Formally, if u is a solution of (I. 1), then the function defined by

(II. 1) v(x, t) = u(x, t) - p(x) p(1) H(t) = u(x, t) -x qµ H(t) is solution of (II. 2)              v t -v xx - µ x 2 v = - p(x) p(1) H (t) x ∈ (0, 1), t ∈ (0, T ), v(0, t) = 0 t ∈ (0, T ), v(1, t) = 0 t ∈ (0, T ), v(x, 0) = u 0 (x) -p(x)
p(1) H(0)

x ∈ (0, 1).

Reciprocally, given h ∈ L 2 (0, T ), consider the solution of

             v t -v xx - µ x 2 v = - p(x) p(1) h(t) x ∈ (0, 1), t ∈ (0, T ), v(0, t) = 0 t ∈ (0, T ), v(1, t) = 0 t ∈ (0, T ), v(x, 0) = v 0 (x)
x ∈ (0, 1).

Then the function u defined by

(II. 3) u(x, t) = v(x, t) + p(x) p(1) t 0 h(τ )dτ satisfies (II. 4)            u t -u xx - µ x 2 u = 0 x ∈ (0, 1), t ∈ (0, T ), u(0, t) = 0 t ∈ (0, T ), u(1, t) = t 0 h(τ )dτ t ∈ (0, T ), u(x, 0) = v 0 (x)
x ∈ (0, 1).

This motivates the fact that we will first establish results of well-posedness for an auxiliary problem with Dirichlet homogeneous boundary conditions and source term (see section II.2). Then we will define the solutions of the boudary value problem (I. 1) via this auxiliary problem (see section II.3).

II.2. Homogeneous boundary conditions and a source term. Let us first consider the system with homogeneous boundary conditions and a source term

(II. 5)            w t -w xx - µ x 2 w = f (x, t) x ∈ (0, 1), t ∈ (0, T ), w(0, t) = 0 t ∈ (0, T ), w(1, t) = 0 t ∈ (0, T ), w(x, 0) = w 0 (x)
x ∈ (0, 1).

We define H 1 0 (µ) the Hilbert space obtained as the closure of H 1 0 (0, 1) with respect to the norm ∀z ∈ H 1 0 (0, 1),

z µ := 1 0 z 2 x -µ z 2 x 2 dx 1/2
.

(Thanks to the Hardy inequality (I. 2), this defines a norm for any value of the paramater µ ≤ 1/4).

In the sub-critical case µ < 1/4, the norm • µ is equivalent to the standard norm of H 1 0 (0, 1) (see [START_REF] Vázquez | The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential[END_REF] p. 115). Therefore, H 1 0 (µ) = H 1 0 (0, 1) for any µ < 1/4. In the critical case µ = 1/4, it has been proved (see [START_REF] Vázquez | The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential[END_REF] p. 127) that H 1 0 (µ = 1/4) is strictly larger that H 1 0 (0, 1):

H 1 0 (0, 1)⊂ = H 1 0 (µ = 1/4).
Observe that, if we denote H 1 (µ) the Hilbert space obtained as the completion of H 1 (0, 1) with respect to the norm z L 2 (0,1) + z µ , we have

H 1 0 (µ) = {z ∈ H 1 (µ) | z(0) = 0 = z(1)}. Let us define the operator L µ : D(L µ ) ⊂ L 2 (0, 1) → L 2 (0, 1) by: D(L µ ) = {z ∈ H 1 0 (µ) | -z xx -µ z x 2 ∈ L 2 (0, 1)}, L µ z = -z xx -µ z x 2 .
We also define

H 2 (µ) = {z ∈ H 1 (µ) | -z xx -µ z x 2 ∈ L 2 (0, 1)} so that D(L µ ) = H 2 (µ) ∩ H 1 0 (µ).
It can be proved that, for any µ ≤ 1/4, L µ is self-adjoint with compact inverse. In particular, we have:

Theorem II.1. (see [START_REF] Vázquez | The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential[END_REF])

Assume µ ≤ 1/4. There exists an orthonormal basis (Φ k ) k≥1 of L 2 (0, 1) constituted of eigenvectors of L µ with eigenvalues sequence

0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ k ≤ . . . → +∞, so that L µ Φ k = λ k Φ k in (0, 1), Φ k (0) = 0 = Φ k (1).
Besides L µ generates an analytic semi-group of contractions in L 2 (0, 1) for the equation (II. 5) (see also [START_REF] Vázquez | The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential[END_REF]). As a consequence, for any w 0 ∈ L 2 (0, 1) and f ∈ L 2 ((0, 1) × (0, T )), problem (II. 5) is well-posed : Definition II.1. a) Given w 0 ∈ L 2 (0, 1) and f ∈ L 2 ((0, 1) × (0, T )), one defines the mild solution of (II. 5)

w ∈ C 0 ([0, T ]; L 2 (0, 1)) ∩ L 2 (0, T ; H 1 0 (µ))
as the one given by the variation formula:

w(x, t) = e tLµ w 0 + t 0 e (t-s)Lµ f (x, s)ds.
b) Moreover, we say that a function

w ∈ C 0 ([0, T ]; H 1 0 (µ)) ∩ H 1 (0, T ; L 2 (0, 1)) ∩ L 2 (0, T ; D(L µ )
) is a strict solution of (II. 5) if it satisfies the equation a.e. in (0, 1) × (0, T ) and the boundary and initial conditions for all t ∈ [0, T ] and x ∈ [0, 1].

Notice that, if w 0 ∈ H 1 0 (µ), then the mild solution of (II. 5) is also the unique strict solution. II.3. Non homogeneous boundary condition. Next we turn to the boundary value problem (I. 1). Let H be given in H 1 (0, T ). The results of section II.2 apply in particular to problem (II. 5) when one chooses

(II. 6) f (x, t) = - p(x) p(1) H (t) and v 0 (x) = u 0 (x) - p(x) p(1) H(0).
This allows us to define in a suitable way the solution of (I. 1):

Definition II.2. a) We say that u ∈ C 0 ([0, T ]; L 2 (0, 1)) ∩ L 2 (0, T ; H 1 (µ))
is the mild solution of (I. 1) if v defined by (II. 1) and (II. 6) is the mild solution of

(II. 2). b) We say that u ∈ C 0 ([0, T ]; H 1 (µ)) ∩ H 1 (0, T ; L 2 (0, 1)) ∩ L 2 (0, T ; H 2 (µ))
is the strict solution of (I. 1) if v defined by (II. 1) and (II. 6) is the strict solution of (II. 2).

We deduce

Proposition II.1. a) Given u 0 ∈ L 2 (0, 1) and H ∈ H 1 (0, T ), problem (I. 1) admits a unique mild solution. b) Given u 0 ∈ H 1 (µ) such that u 0 (0) = 0 and H ∈ H 1 (0, T ) such that H(0) = u 0 (0), problem (I. 1
) admits a unique strict solution. (In particular, this holds true when

u 0 ∈ H 1 0 (µ), H ∈ H 1 (0, T ) such that H(0) = 0). The proof of Proposition II.1 follows immediately noting that H(x, t) := p(x) p(1) H(t) satisfies H ∈ C 0 ([0, T ]; H 1 (µ)) ∩ H 1 (0, T ; L 2 (0, 1)) ∩ L 2 (0, T ; H 2 (µ)).
Remark II.1. In the case u 0 ∈ L 2 (0, 1), we have :

for any τ > 0, v(•, τ ) ∈ H 2 (µ) ∩ H 1 0 (µ). Therefore the solution of (II. 2) is strict on [τ, T ].
The same is true for the solution of (I. 1).

III. Proof of the main results (Theorems I.1 and I.2)

In this section, we prove our main result. In order to straight the main lines of the proof, some computations are postponed in the technical part in section IV. III.1. Eigenvalues and eigenbasis of the Sturm-Liouville Problem. In order to transform the question of null controllability into a moment problem, let us first determine the eigenfunctions and eigenvalues associated to the operator L µ . This means that we aim at solving the following boundary value problem for any suitable λ ∈ R:

(III. 1) -φ (x) - µ x 2 φ(x) = λφ(x)
x ∈ (0, 1),

φ(0) = 0 = φ(1).
As this problem is closely related to Bessel's equations, we will need to use throughout this paper some standard definitions, notations and properties coming from Bessel's theory. For reader convenience, we recall and summarize the elements that are useful in the appendix in section V.

Concerning problem (III. 1), we prove (see section IV.1 for the proof):

Proposition III.1. Assume µ ≤ 1/4 and define

ν(µ) := 1 4 -µ.
We denote by J ν(µ) the Bessel function of first kind of order ν(µ) (see section V) and we denote 0

< j ν,1 < j ν,2 < • • • < j ν,n < • • • → +∞ as n → +∞ the sequence of positive zeros of J ν(µ)
. Then the admissible eigenvalues λ for problem (III. 1) are ∀n ≥ 1, λ µ,n = (j ν(µ),n ) 2 and the corresponding (normalized) eigenfunctions are

∀n ≥ 1, Φ µ,n (x) = 1 |J ν(µ) (j ν(µ),n )| √ xJ ν(µ) (j ν(µ),n x),
x ∈ (0, 1).

Moreover the family (Φ µ,n ) n≥1 forms an orthonormal basis of L 2 (0, 1).

III.2. Useful estimates on eigenvalues and eigenfunctions.

Next we give some estimates on the eigenvalues (proved later in section IV.2) that will be useful in the analysis of the problem:

Lemma III.1. (i) When µ ∈ [0, 1/4], then (III. 2) ∀n ≥ 1, 7π 8 ≤ λ µ,n+1 -λ µ,n ≤ π.
(ii) When µ ≤ 0, then

(III. 3) ∀n ≥ 1, π ≤ λ µ,n+1 -λ µ,n , and
(III. 4) ∀n > ν(µ), λ µ,n+1 -λ µ,n ≤ 2π.
Observe that (III. 2) and (III. 3) ensure a lower estimate of the gap that is uniform for any µ ≤ 1/4:

∀µ ∈ [0, 1/4], ∀n ≥ 1, 7π 8 ≤ λ µ,n+1 -λ µ,n .
This will enable us to use the standard methods developped by Fattorini-Russell [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] to give an upper estimate of the cost. More precisely, we use a result proved in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] that makes explicit the one obtained in [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF] (in short time). For this part, we will able to treat the two cases µ ∈ [0, 1/4] and µ ≤ 0 in the same way.

As for the obtention of a lower estimate of the controllability cost, we will have to treat separately the two cases µ ∈ [0, 1/4] and µ ≤ 0, using extensions of Guichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF], that we proved in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] and [START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF]. III.3. Transformation into a moment problem. Following the strategy initiated by Fattorini and Russell [START_REF] Fattorini | Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension[END_REF][START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF], we reduce here the controllability question to some moment problem. In this part, we analyze the problem with formal computations.

First, we expand the initial condition u 0 ∈ L 2 (0, 1): there exists (

β 0 µ,n ) n≥1 ∈ 2 (N ) such that u 0 (x) = n≥1 β 0 µ,n Φ µ,n (x).
Next we expand the solution u of (I. 1):

u(x, t) = n≥1 β µ,n (t)Φ µ,n (x), x ∈ (0, 1), t ≥ 0 with n≥1 β µ,n (t) 2 < +∞.
Therefore the controllability condition u(•, T ) = 0 becomes

∀n ≥ 1, β µ,n (T ) = 0.
On the other hand, we observe that w µ,n (x, t) := Φ µ,n (x)e λµ,n(t-T ) is solution of the adjoint problem:

(III. 5)      (w µ,n ) t + (w µ,n ) xx + µ x 2 w µ,n = 0 x ∈ (0, 1), t > 0, w µ,n (0, t) = 0 t > 0 w µ,n (1, t) = 0 t > 0.
Multiplying (I. 1) by w µ,n and (III. 5) by u, we obtain

1 0 uwdx T 0 - T 0 [u x w] 1 0 dt + T 0 [w x u] 1 0 dt = 0.
Taking into account that u(T ) ≡ 0 and the boundary conditions satisfied by u and w, we obtain

- 1 0 u 0 Φ µ,n e -λµ,nT dx + T 0 Φ µ,n (1)
e λµ,n(t-T ) H(t) = 0.

Therefore the question reduces into the following moment problem : find H such that

(III. 6) ∀n ≥ 1, r µ,n T 0 H(t)e λµ,nt dt = β 0 µ,n ,
where we set r µ,n = Φ µ,n (x = 1).

As we look for a control H belonging to H 1 (0, T ) such that H(0) = 0 = H(T ), we rather write the problem satisfied by H (t) which is simply obtained by some integration by part in time:

(III. 7) ∀n ≥ 1, - r µ,n λ µ,n T 0 H (t)e λµ,nt dt = β 0 µ,n .
III.4. Formal solution of the moment problem. Set artificially λ µ,0 := 0, and assume for a moment that we are able to construct a family (σ + µ,m ) m≥0 of functions σ + µ,m ∈ L 2 (0, T ), which is biorthogonal to the family (e λµ,nt ) n≥0 , which means that:

(III. 8) ∀m, n ≥ 0, T 0 σ + µ,m (t)e λµ,nt dt = δ mn = 1 if m = n, 0 if m = n.
Then let us define (III. 9)

K(t) = - +∞ m=1 λ µ,n β 0 µ,n r µ,n σ + µ,m (t)
and

H(t) = t 0 K(τ )dτ.
Then it is easy to show that, at least formally, K solves the moment problem for the derivative (III. 7). Moreover, if K ∈ L 2 (0, T ), then clearly H ∈ H 1 (0, T ), H = K, and H(0) = 0. Moreover H(T ) = 0 thanks to the additional property that the family (σ + µ,m ) m≥1 is orthogonal to e λµ,0t = 1. So H will be in H 1 (0, T ) such that H(0) = 0 = H(T ) and will satisfy the moment problem (III. 6).

It remains to check that all this makes sense. For this purpose, we will have to prove the existence of a biortogonal family (σ + µ,m ) m≥0 together with suitable L 2 bounds (so that we can prove that K ∈ L 2 (0, T )). III.5. Existence of a suitable biorthogonal family. We will use the following result:

Theorem III.1. (see Theorem 2.4 in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF]) Assume that ∀n ≥ 0, λ n ≥ 0, and that there is some γ min > 0 such that

(III. 10) ∀n ≥ 0, λ n+1 -λ n ≥ γ min .
Then there exists a family (σ + m ) m≥0 which is biorthogonal to the family (e λnt ) n≥0 in L 2 (0, T ):

(III. 11) ∀m, n ≥ 0, T 0 σ + m (t)e λnt dt = δ mn .
Moreover, it satisfies: there is some universal constant C u independent of T , γ min and m such that, for all m ≥ 0, we have

(III. 12) σ + m 2 L 2 (0,T ) ≤ C u e -2λmT e
B(T, γ min ) =    1 T + 1 T 2 γ 2 min e Cu γ 2 min T if T ≤ 1 γ 2 min , C u γ 2 min if T ≥ 1 γ 2 min ,
and this is clearly equivalent to (III. 12)-(III. 13). Its proof is based on complex analysis tools, combining the approach of Seidman-Avdonin-Ivanov [START_REF] Seidman | The "window problem" for series of complex exponentials[END_REF] with the addition of some parameter, as in Tucsnak-Tenenbaum [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrodinger and heat equations[END_REF].

As we have already noted (see Lemma III.1), the eigenvalues of the problem satisfy for all µ ≤ 1/4:

∀n ≥ 1, λ µ,n+1 -λ µ,n ≥ 7π/8.
Define artificially λ µ,0 := 0.

Then, for all µ ≤ 1/4,

λ µ,1 -λ µ,0 = j ν(µ),1 ≥ 3π/4,
using the fact that, thanks to (V. 5) and (V. 6), one can easily prove that j ν,1 ≥ 3π/4 for all ν ≥ 0. Therefore we can apply Theorem III.1 to the family (e λµ,nt ) n≥0 provided that we choose γ min = min(7π/8, 3π/4) = 3π/4. We obtain that there exists a family (σ + µ,m ) m≥0 biorthogonal to (e λµ,nt ) n≥0 in L 2 (0, T ), and such that

σ + µ,m 2
L 2 (0,T ) ≤ Ce -2λµ,mT e C √ λµ,m B(T ) with B(T ) = max 1, 1
T 2 e C/T for all T > 0.

III.6. Upper bounds of the cost: proof of (I. 4) and (I. 6). Let us first state the preliminary Lemma (see section IV.3 for its proof):

Lemma III.2. For all µ ≤ 1/4 and all n ≥ 1, r µ,n = (-1) n j ν(µ),n .

Then define (III. [START_REF] Fotouhi | Null controllability of degenerate/singular parabolic equations[END_REF])

K(t) := - ∞ m=1 λ µ,m β 0 µ,m r µ,m σ + µ,m (t), and
H(t) := t 0 K(τ ) dτ,
and let us check that H is an admissible control that drives the solution of (I. 1) to 0 in time T :

• first we check that K ∈ L 2 (0, T ): let us write ∞ m=1 |λ µ,m β 0 µ,m | |r µ,m | σ + µ,m L 2 (0,T ) ≤ ∞ m=1 |β 0 µ,m | 2 1/2 ∞ m=1 |λ µ,m | 2 |r µ,m | 2 σ + µ,m 2 L 2 (0,T ) 1/2 . Since |r µ,m | 2 = (j ν(µ),m ) 2 = λ µ,m , it follows that ∞ m=1 |λ µ,m β 0 µ,m | |r µ,m | σ + µ,m L 2 (0,T ) ≤ C u 0 L 2 (0,1) ∞ m=1 λ µ,m e -2λµ,mT e C √ λµ,m B(T ) 1/2
which is finite. This implies that K ∈ L 2 (0, T ). Therefore we have H ∈ H 1 (0, T ) with of course H(0) = 0. And the fact that H(T ) = 0 follows from (III. 11) with n = 0;

• next, we check that H = K satisfies the moment problem (III. 7):

∀n ≥ 1, - r µ,n λ µ,n T 0 H (t)e λµ,nt dt = r µ,n λ µ,n T 0 ∞ m=1 λ µ,m β 0 µ,m r µ,m σ + µ,m (t) e λµ,nt dt = r µ,n λ µ,n ∞ m=1 λ µ,m β 0 µ,m r µ,m δ mn = β 0 µ,n ;
• finally we check that the solution of (I. 1) satisfies u(T ) = 0: multiplying the first equation of (I. 1) by w µ,n (x, t) := Φ µ,n (x)e λµ,n(t-T ) and integrating by parts, we obtain that

∀n ≥ 1, 1 0 u(x, T )Φ µ,n (x) dx = 0,
hence u(T ) ≡ 0. Therefore H is an admissible control, and it follows that

C ctr-bd ≤ H H 1 (0,T ) u 0 L 2 (0,1) ≤ C K L 2 (0,T ) u 0 L 2 (0,1) , hence C ctr-bd ≤ C ∞ m=1 |λ µ,m | 2 |r µ,m | 2 σ + µ,m 2 L 2 (0,T ) 1/2 ≤ C B(T ) ∞ m=1 λ µ,m e -2λµ,mT e C √ λµ,m 1/2 . But λ µ,m = (j ν(µ),m ) 2 and C λ µ,m ≤ λ µ,m T + C T = j 2 ν(µ),m T + C T .
One deduces that

C ctr-bd ≤ C B(T )e C /T ∞ m=1 (j ν(µ),m ) 2 e -j 2 ν(µ),m T 1/2 ≤ Ce C"/T ∞ m=1 (j ν(µ),m ) 2 e -j 2 ν(µ),m T 1/2 .
Next we use the following Lemma:

Lemma III.3. There is some constant (independant of ν and of Y ) such that :

(III. 17) ∀ν ≥ 0, ∀T > 0, ∞ m=1 (j ν,m ) 2 e -j 2 ν,m T ≤ C 1 + ν 2 T 3/2 e -(1+ν 2 )T /C .
Proof of Lemma III.3. It is based on classical analysis estimates, see [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF].

Applying Lemma III.3, it follows that:

C ctr-bd ≤ Ce C"/T 1 + ν T 3/4 e -(1+ν 2 )T /(2C) , which implies (I. 4) and (I. 6). III.7. Lower bound of the cost when µ ∈ [0, 1 4 ]: proof of (I. 5). Given m ≥ 1, consider u 0 = Φ µ,m , and let H m be any control that drives the solution of (I. 1) to 0 in time T . Then (III. 6) gives that ∀n ≥ 1, T 0 r µ,m H m (t) e λµ,nt dt = δ mn .

Hence the sequence (r µ,m H m ) m≥1 is biorthogonal to the set (e λµ,nt ) n≥1 . There exist several lower bounds in the literature for such biorthogonal sequences, in particular Guichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] and Hansen [START_REF] Hansen | Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems[END_REF]. In this case we are going to use the following generalization of Guichal [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF], proved in [START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF]:

Theorem III.2. (Theorem 2.5 in [6]) Assume that ∀n ≥ 1, λ n ≥ 0,
and that there is some 0 < γ min ≤ γ max such that

(III. 18) ∀n ≥ 1, γ min ≤ λ n+1 -λ n ≤ γ max .
Then there exists c u > 0 independent of T , and m such that: any family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ) satisfies: 2 . and

(III. 19) σ + m 2 L 2 (0,T ) ≥ e -2λmT e 1 2γ 2 max T b(T, γ max , m), with (III. 20) b(T, γ max , m) = c 2 u C(m, γ max , λ 1 ) 2 T ( 1 2γ 2 max T ) 2m 1 (4γ 2 max T + 1)
(III. 21) C(m, γ max , λ 1 ) = m! 2 m+[ 2 √ λ 1 γmax ]+1 (m + [ 2 √ λ 1 γ max ] + 1).
(The proof of Theorem III.2 is a natural generalization of the Hilbertian techniques used in [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF].) When µ ∈ [0, 1 4 ], we are in position to apply Theorem III.2. Indeed, using (III. 2), we see that the assumption (III. 18) is satisfied with

γ min := 7π 8
, and γ max := π.

We are then in position to apply Theorem III.2, and we obtain that any family (σ + µ,m ) m≥1 which is biorthogonal to the family (e λµ,nt ) n≥1 in L 2 (0, T ) satisfies:

(III. 22) σ + µ,m 2
L 2 (0,T ) ≥ e -2λµ,mT e 32 81π 2 T b(T, γ max , m),
with the expression of b(T, γ max , m) given in (III. 20).

In the following, we apply this inequality for m = 1. Observe that, for ν ∈ [0, 1/2] and n = 1, (V. 5) gives

3π 4 ≤ π 3 4 + ν 2 ≤ j ν,1 ≤ π 1 + 1 4 ν - 1 2 ≤ π.
Hence (3π/4) 2 ≤ λ µ,1 ≤ π 2 , and λ µ,1 ≤ C(1 + ν(µ)).

In particular, choosing m = 1 in (III. 22) and using λ µ,1 ≥ (3π/4) 2 , we obtain that there exists c u independent of T > 0 and µ ∈ [0, 1 4 ) such that b(T, γ max , 1) ≥ c u T 3 (1 + T ) 2 . III.8. Lower bound of the cost when µ ≤ 0: proof of (I. 7). In this case, one still could apply Theorem III.2 but this would not give a good result. Indeed, in this case we have

Next we deduce

σ + µ,1 2 L 2 (0,T ) ≥ c u T 3 (1 + T ) 2 e -2λµ,1T e 32 81π 2 T . Hence r µ,1 H 1 2 L 2 (0,T ) ≥ c u T 3 (1 + T ) 2 e -2λµ,
λ µ,n+1 -λ µ,n = j ν(µ),n+1 -j ν(µ),n ,
and since µ ≤ 0, we have ν(µ) ≥ 1 2 . Hence we deduce from Komornik-Loreti [START_REF] Komornik | Fourier Series in Control Theory[END_REF] p. 135 that the sequence (j ν(µ),n+1 -j ν(µ),n ) n≥1 is nonincreasing and converges to π, hence π ≤ j ν(µ),n+1 -j ν(µ),n ≤ j ν(µ),2 -j ν(µ),1 .

This says that assumption (III. 18) is satisfied, and that the best values for γ min and γ max are:

γ min = π, γ max = j ν(µ),2 -j ν(µ),1 .
But it follows from (V. 7) that there exists a > 0 such that

j ν,2 -j ν,1 ∼ a ν 1/3 as ν → +∞. Hence j ν(µ),2 -j ν(µ),1 ∼ aν(µ) 1/3 → +∞ as µ → -∞.
Therefore applying Theorem III.2 does not give a good result (since now γ max → +∞ as µ → -∞).

On the other hand, from point (iii) of Lemma III.

Theorem III.3. (Theorem 2.2 in [7]) Assume that ∀n ≥ 1, λ n ≥ 0,
and that there are

0 < γ min ≤ γ * max ≤ γ max such that (III. 25) ∀n ≥ 1, γ min ≤ λ n+1 -λ n ≤ γ max , and
(III. 26) ∀n ≥ N * , λ n+1 -λ n ≤ γ * max .
Then any family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ) satisfies:

(III. 27) σ + m 2 L 2 (0,T ) ≥ e -2λmT e 2 T (γ * max ) 2 b * (T, γ max , γ * max , N * , λ 1 , m) 2
, where b * is rational in T (and explictly given in Lemma 4.4 of [START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF]).

(The proof of Theorem III.3 is a natural generalization of the Hilbertian techniques used in [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF]; Theorem III.3 was motivated by several problems where it appears that the global gap is quite bad, while there is a much better asymptotic gap, see [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF] for an application to degenerate parabolic equations.)

We are in position to apply Theorem III.3: indeed, (III. 25) is satisfied with γ min := π, and γ max := j ν(µ),2 -j ν(µ),1 , and (III. 26) is satisfied choosing N * and γ * max as in (III. 24). Then, applying Theorem III.3, we obtain that any family (σ + µ,m ) m≥1 which is biorthogonal to the family (e λµ,nt ) n≥1 in L 2 (0, T ) satisfies

σ + µ,m 2 L 2 (0,T ) ≥ e -2λµ,mT e 2 4π 2 T b * (T, γ max , γ * max , N * , λ µ,1 , m) 2
with an explicit value of b * (see Lemma 4.4 in [START_REF] Cannarsa | Precise estimates for biorthogonal families under asymptotic gap conditions[END_REF]): when m ≤ N * , we have (III. 28)

b * (T, γ max , γ * max , N * , λ 1 , m) = C * √ 1 + T λ 1 √ T (T (γ * max ) 2 ) K * +K * +2 (1 + (T (γ * max ) 2 )) N * +K * +K * +3
, where

K * = [ 2 √ λ 1 + (N * + m)γ max γ * max ] -N * + 2, K * = [ γ max γ * max (N * -m)] -N * + 2, C * = 1 (N * + K * + K * + 3)! c u (γ * max ) 2(N * -1) C (+) C (-) ,
where

C (+) = ( γ max γ * max ) N * -1 (N * + m + [ 2 √ λ1 γmax ] + 1)! (m + [ 2 √ λ1 γmax ] + 1)! ([ 2 √ λ1+(N * +m)γmax γ * max ] + 1)! (2m + [ 2 √ λ1 γmax ] + 1)
, and

C (-) = ( γ max γ * max ) N * -1 (m -1)! (N * -m)! (1 + [ γmax γ * max (N * -m)])! .
These expressions seem be a little frightening, but we are looking for the behavior as µ → -∞ i.e. ν(µ) → +∞, and this is not difficult to study (see [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF]), and to obtain that, when m = 1:

b * (T, γ max , γ * max , N * , λ µ,1 , 1) ≥ e -Cν(µ) 4/3 (ln ν(µ)+ln 1 T ) √ 1 + T √ T , hence σ + µ,1 2
L 2 (0,T ) ≥ b(T, µ, 1) 2 , with (III. 29) b(T, µ, 1) := e -λµ,1T e 1 4π 2 T √ 1 + T √ T e -Cν(µ) 4/3 (ln ν(µ)+ln 1 T ) .
This gives the following lower bound of the cost when µ ≤ 0:

C ctr-bd ≥ 1 |r µ,1 | b(T, µ, 1) = 1 j ν(µ),1 b(T, µ , 1)
So, using (V. 6), we get

C ctr-bd ≥ 8 π(7 + 2ν(µ))
e -λµ,1T e 1 4π 2 T e -Cν(µ) 4/3 (ln ν(µ)+ln 1

T ) √ 1 + T √ T ≥ e -π 2 ( 3 4 + ν(µ) 2 ) 2 T e 1 4π 2 T e -C ν(µ) 4/3 (ln ν(µ)+ln 1 T ) √ 1 + T √ T ≥ e -π 2 2 ν(µ) 2 T e -9π 2 8 T e 1 4π 2 T e -C ν(µ) 4/3 (ln ν(µ)+ln 1 T ) .
This proves (I. 7).

IV. Technical part

IV.1. Proof of Proposition III.1. Let us first observe that any admissible eigenvalue λ satisfies λ > 0 (use for instance Theorem II.1). Therefore in the following we assume λ = 0. Using the following changes of variables,

φ(x) = √ xψ( √ λx) and y = √ λx,
one can easily see that φ satisfies (III. 1) if and only if ψ is solution of (IV. 1)

   y 2 ψ (y) + yψ (y) + y 2 - 1 4 -µ ψ(y) = 0 y ∈ (0, √ λ), ψ(0) = 0 = ψ( √ λ).
Hence ψ is a solution of the Bessel equation (V. 1) (see section V.1) of order

ν(µ) := 1 4 -µ.
Let us now solve (IV. 1).

IV.1.a. Case 1 : ν(µ) ∈ N. Let us first treat the case ν(µ) ∈ N. Observe that, in that case, µ = 1/4 so H 1 0 (µ) = H 1 0 (0, 1). The space of solutions of the differential equation (IV. 2) y 2 ψ (y) + yψ (y) + (y 2 -ν(µ))ψ(y) = 0 is a vector space of dimension 2. Since ν(µ) ∈ N, a fundamental system of solutions of (IV. 2) is given by the Bessel's functions of the first kind : J ν(µ) and J -ν(µ) , (see section V.2). Hence solutions of equation (IV. 2) are linear combinations of J ν(µ) and J -ν(µ) :

∀y ∈ (0, √ λ), Ψ(y) = C + J ν(µ) (y) + C -J -ν(µ) (y), with C + , C -∈ R. Thus ∀x ∈ (0, 1), Φ(x) = C + √ xJ ν(µ) ( √ λx) + C - √ xJ -ν(µ) ( √ λx).
We will denote

(IV. 3) Φ + (x) := √ xJ ν(µ) ( √ λx), Φ -(x) := √ xJ -ν(µ) ( √ λx).
Using the development in series of J ν(µ) and J -ν(µ) (see section V.2), we obtain:

Φ + (x) = ∞ m=0 c + ν(µ),m √ λ 2m+ν(µ)
x 2m+ν(µ)+1/2 , (IV. 4)

Φ -(x) = ∞ m=0 c - ν(µ),m √ λ 2m-ν(µ)
x 2m-ν(µ)+1/2 . (IV. 5)

We will denote

(IV. 6) c+ ν(µ),m := c + ν(µ),m √ λ 2m+ν(µ) , c- ν(µ),m := c - ν(µ),m √ λ 2m-ν(µ)
, in such a way that

(IV. 7) Φ + (x) = ∞ m=0 c+ ν(µ),m x 2m+ν(µ)+1/2 , Φ -(x) = ∞ m=0 c- ν(µ),m x 2m-ν(µ)+1/2 .
Then, let us verify that Φ + ∈ H 1 (0, 1):

(IV. 8) Φ + (x) 2 ∼ x→0 c+ ν(µ),0 2 x 2ν(µ)+1 → x→0 0. So we have Φ + ∈ L 2 (0, 1). Next Φ + (x) 2 ∼ x→0 c+ ν(µ),0 2 (ν(µ) + 1/2) 2
x 1-2ν(µ) .

Since ν(µ) > 0, we have 1 -2ν(µ) < 1 and Φ + ∈ L 2 (0, 1). Hence Φ + ∈ H 1 (0, 1).

Next we look at Φ -:

Φ -(x) 2 ∼ x→0 c- ν(µ),0 2
x 2ν(µ)-1 . We deduce that Φ -∈ L 2 (0, 1) if only if ν(µ) < 1 i.e. µ > -3/4. Moreover, even in the case µ > -3/4, we have Φ -∈ L 2 (0, 1) since

Φ -(x) 2 ∼ 0 ∼ x→0 c- ν(µ),0 2 (-ν(µ) + 1/2) 2 x 2ν(µ)+1
with ν(µ) > 0. So Φ -never belongs to H 1 (0, 1).

In conclusion, we have

Φ + ∈ H 1 (0, 1) but Φ -∈ H 1 (0, 1). It implies that C -= 0 and so Φ(x) = C + Φ + (x) = C + √ xJ ν(µ) ( √ λx).
Now let us take into account the boundary conditions. From (IV. 8), we have Φ + (0) = 0 so the condition Φ(0) = 0 is automatically satisfied. Next the condition Φ(1) = 0 turns into C + J ν(µ) ( √ λ) = 0 which means that, in order to get a non trivial solution, √ λ has to be a zero of J ν(µ) . So there exists some n such that

λ = (j ν(µ),n ) 2 .
Finally, ∀x ∈ (0, 1),

Φ(x) = C + √ xJ ν(µ) (j ν(µ),n x).
Reciprocally, for all n ≥ 1 and all C + ∈ R, we have a solution of (IV. 1). Hence we have solved (IV. 1). Now consider

∀n ≥ 1, Φµ,n (x) := √ xJ ν(µ) (j ν(µ),n x).
The orthogonality properties of the Bessel's functions imply that (Φ µ,n ) n forms an orthogonal family of L 2 (0, 1): indeed, using (V. 8) in section V.5, we have

1 0 Φµ,n (x) Φµ,m (x) dx = 1 0 xJ ν(µ) (j ν(µ),n x) J ν(µ) (j ν(µ),m x) dx = δ nm [J ν(µ)+1 (j ν(µ),n )] 2 = δ nm [J ν(µ) (j ν(µ),n )] 2 ,
where we also used the fact that Bessel function of the first kind satisfy the following recurrence formulae (see [44, p. 45, relation (4)]):

xJ ν (x) -νJ ν (x) = -xJ ν+1 (x), hence J ν+1 (j ν,n ) = -J ν (j ν,n ).
Therefore one can normalize the eigenfunctions Φµ,n into

Φ µ,n (x) = 1 |J ν(µ) (j ν(µ),n )| √ xJ ν(µ) (j ν(µ),n x), x ∈ (0, 1), n ≥ 1.
Finally the family (Φ µ,n ) n≥1 forms an orthonormal basis of L 2 (0, 1) since they are the eigenfunctions of the operator L µ .

IV.1.b. Case 2 : ν(µ) = n(µ) ∈ N * . Let us assume that ν(µ) = n(µ) ∈ N * . Observe that, once again in that case, µ = 1/4 so H 1 0 (µ) = H 1 0 (0, 1). Since ν(µ) ∈ N * , a fundamental system of solutions of (IV. 2) is given by the Bessel's functions of the first kind and the Bessel's functions of the second kind : J ν(µ) and Y ν(µ) , (see section V.3). So, if we denote

(IV. 9) Φ +,-(x) := √ xY n(µ) ( √ λx),
then Φ takes the form Φ = C + Φ + + C +,-Φ +,-. We proved in case 1 that Φ + ∈ H 1 (0, 1). It remains to study if Φ +,-∈ H 1 (0, 1). Using the decomposition in series of Y n(µ) (see (V. 4) in section V.3), we get (IV. 10)

Φ +,-(x) = 2 π Φ + (x) log √ λx 2 + n(µ)-1 m=0 âm x 2m-n(µ)+1/2 + +∞ m=0 bm x n(µ)+2m+1/2 ,
where In the following, we study these three functions that appear in the formula of Φ +,-. First, let us study

âm := - 1 π (n(µ) -m -1)! m! √ λ 2 2m-n(µ)
(IV. 11) Φ +,-,1 (x) := 2 π Φ + (x) ln √ λx 2 .
It satisfies

Φ +,-,1 (x) ∼ x→0 2 π c+ n(µ),0 x n(µ)+1/2 ln √ λx 2 .
Hence we deduce that Φ +,-,1 ∈ L 2 (0, 1). Moreover

Φ +,-,1 (x) = 2 π Φ + (x) ln √ λx 2 + 2 π Φ + (x) 1 x ∼ x→0 2 π c+ n(µ),0 (n(µ) + 1/2) x 1/2-n(µ) ln √ λx 2 .
Since n(µ) ≥ 1, we deduce that Φ +,-,1 ∈ L 2 (0, 1) and finally Φ +,-,1 ∈ H 1 (0, 1). Next we study

Φ +,-,2 (x) := n(µ)-1 m=0 âm x 2m-n(µ)+1/2 .
It satisfies Φ +,-,2 (x) ∼ 0 â0 x -n(µ)+1/2 , Φ +,-,2 (x) ∼ 0 â0 (-n(µ) + 1/2)x -n(µ)-1/2 , hence Φ +,-,2 / ∈ H 1 (0, 1), since â0 = 0 and n(µ) > 0.

Finally, let us observe that (IV. 12) Φ +,-,3 (x) := +∞ m=0 bm x n(µ)+2m+1/2 satisfies Φ +,-,3 (x) ∼ 0 b0 x n(µ)+1/2 , Φ +,-,3 (x) ∼ 0 b0 (n(µ) + 1/2)x n(µ)-1/2 .

Since n(µ) ≥ 1, we deduce that Φ +,-,3 ∈ H 1 (0, 1). Finally Φ +,-= Φ +,-,1 +Φ +,-,2 +Φ +,-,3 / ∈ H 1 (0, 1). Consequently Φ = C + Φ + + C +,-Φ +,-∈ H 1 (0, 1) necessarily implies that C +,-= 0. Then Φ = C + Φ + and we are in the same position as in case 1 and the conclusion is the same.

IV.1.c. Case 3 : ν(µ) = 0 (i.e. µ = 1/4). Let us assume that ν(µ) = 0. Observe that, in that case, µ = 1/4 so H 1 0 (1/4) = H 1 0 (0, 1). We recall that

H 1 0 (1/4) = {z ∈ H 1 (1/4) | z(0) = 0 = z(1)}
where H 1 (1/4) is the Hilbert space obtained as the completion of H 1 (0, 1) with respect to the norm z L 2 (0,1) + z 1/4 with

z 1/4 := 1 0 z 2 x - 1 4 z 2 x 2 dx 1/2 .
As in case 2, since ν(1/4) = 0 ∈ N, Φ takes the form Φ = C + Φ + + C +,-Φ +,-.

In the following, we will first prove that Φ + ∈ H 1 (1/4) whereas Φ +,-∈ H 1 (1/4). This implies that Φ = C + Φ + otherwise Φ ∈ H 1 (1/4). Then we take into account the boundary conditions coming from the fact that Φ ∈ H 1 0 (1/4) and the conclusion is the same as in case 1.

So let us first prove that Φ + ∈ H 1 (1/4). Taking ν(µ) = 0 in (IV. 7), we get (IV. 13) Φ + (x) = c+ 0,0 x 1/2 + c+ 0,1 x 5/2 + o(x 5/2 ) = c+ 0,0 x 1/2 1 + c+ 0,1 c+ 0,0

x 2 + o(x 2 ) as x → 0, and also

(IV. 14) Φ + (x) = c+ 0,0 2 x -1/2 + 5c + 0,1 2 x 3/2 + o(x 3/2 ) = c+ 0,0 2 x -1/2 1 + 5c + 0,1 c+ 0,0 x 2 + o(x 2 ) as x → 0. It follows that Φ + (x) 2 - 1 4 Φ + (x) 2 x 2 ∼ x→0 2c + 0,0 c+ 0,1 x. Hence Φ + ∈ H 1 (1/4).
Next, let us prove that Φ +,-∈ H 1 (1/4). In this case, the first sum in the decomposition of Y 0 has to be taken equal to zero (see section V.3 in appendix) , hence we have Φ +,-= Φ +,-,1 + Φ +,-,3 . As in case 2, we study separately the two functions Φ +,-,1 and Φ +,-,3 .

From (IV. 11) and (IV. 13), we get

(IV. 15) Φ +,-,1 (x) = 2 π c+ 0,0 x 1/2 ln √ λx 2 1 + c+ 0,1 c+ 0,0 x 2 + o(x 2 ) as x → 0.
And from (IV. 13) and (IV. 14), we compute

(IV. 16) Φ +,-,1 (x) = c+ 0,0 π x -1/2 ln √ λx 2   1 + 2 ln √ λx 2 + o(x 3/2 )   as x → 0. It follows that Φ +,-,1 (x) 2 - 1 4 Φ +,-,1 (x) 2 x 2 = (c + 0,0 ) 2 π 2 x ln √ λx 2 2   4 ln √ λx 2 + +o(x 3/2 )   ∼ 0 4(c + 0,0 ) 2 π 2 1 x ln √ λx 2 ∈ L 1 (0, 1). Hence Φ +,-,1 ∈ H 1 (1/4).
Next we prove that Φ +,-,3 ∈ H 1 (1/4). Indeed we can compute from (IV. 12):

Φ +,-,3 (x) = b0 x 1/2 1 + b1 b0 x 2 + o(x 2 ) , Φ +,-,3 (x) = 1 2 b0 x -1/2 1 + 5 b1 b0 x 2 + o(x 2 ) . Hence Φ +,-,3 (x) 2 - 1 4 Φ +,-,3 (x) 2 x 2 ∼ 0 2 b0 b1 x ∈ L 1 (0, 1).
Finally this implies that Φ +,-∈ H 1 (1/4) which ends the proof.

IV.2. Proof of Lemma III.1. The proof of Lemma III.1 directly follows from the following one:

Lemma IV.1.

(i) For any ν ∈ [0, 1/2] and any n ≥ 1, 7π/8 ≤ j ν,n+1 -j ν,n ≤ π.

(ii) For any ν ≥ 1/2 and any n ≥ 1, π ≤ j ν,n+1 -j ν,n .

(iii) For any ν ≥ 1/2 and any n > ν, j ν,n+1 -j ν,n ≤ 2π.

Indeed, using that λ µ,n = (j ν(µ),n ) 2 , we get Lemma III.1. So it remains to prove Lemma IV.1. (i) First consider ν ∈ [0, 1 2 ]. Using (V. 5), we get

j ν,n+1 -j ν,n ≥ π( 7 8 + ν 4
).

Since ν ≥ 0, it follows that

j ν,n+1 -j ν,n ≥ 7π 8 .
Moreover it is classical (see [START_REF] Komornik | Fourier Series in Control Theory[END_REF] p. 135) that, when ν ∈ [0, 1 2 ], the sequence (j ν,n+1j ν,n ) n≥1 ) is nondecreasing and converges to π when n → ∞. This completes the proof of (i) of Lemma III.1. (ii) Next consider ν ≥ 1/2. Here we use the fact (see section V.4) that, for ν ≥ 1 2 , the sequence (j ν,n+1 -j ν,n ) n is nonincreasing and converges to π. This ensures that j ν,n+1 -j ν,n ≥ π. (iii) Finally, (iii) is proved in Lemma 5.1 of [START_REF] Cannarsa | The cost of controlling strongly degenerate parabolic equations[END_REF] (in the spirit of Komornik-Loreti [START_REF] Komornik | Fourier Series in Control Theory[END_REF] p. 135, using classical Sturm theory for second order differential equations). IV.3. Proof of Lemma III.2. We compute

Φ µ,n (x) = J ν(µ) (j ν(µ),n x) 2|J ν(µ) (j ν(µ),n )| √ x + √ xJ ν(µ) (j ν(µ),n x)j ν(µ),n |J ν(µ) (j ν(µ),n )| . Hence r µ,n = Φ µ,n (1) = J ν(µ) (j ν(µ),n )j ν(µ),n |J ν(µ) (j ν(µ),n )| = (-1) n j ν(µ),n .
V. Appendix : elements of Bessel Theory

For reader convenience, we recall here the definitions concerning Bessel's equations and functions together with some useful properties of these functions and of their zeros. Throughout this section, we assume that ν ∈ R + .

V.1. Bessel's equation and Bessel's functions of order ν. The Bessel's functions of order ν are the solutions of the following differential equation (see [44, section 3.1, eq. ( 1), p. 38] or [27, eq (5.1.1), p. 98]):

(V. 1) x 2 y (x) + xy (x) + (x 2 -ν 2 )y(x) = 0, x ∈ (0, +∞).
The above equation is called Bessel's equation for functions of order ν. Of course the fundamental theory of ordinary differential equations says that the solutions of (V. 1) generate a vector space S ν of dimension 2. In the following we recall what can be chosen as a basis of S ν .

V.2. Fundamental solutions of Bessel's equation when ν / ∈ N. Assume that ν / ∈ N. When looking for solutions of (V. 1) of the form of series of ascending powers of x, one can construct two series that are solutions: When ν ∈ N, the two functions J ν and J -ν are linearly independent and therefore the pair (J ν , J -ν ) forms a fundamental system of solutions of (V. 1), (see [44, section 3.12, eq. ( 2), p. 43]).

V.3. Fundamental solutions of Bessel's equation when ν = n ∈ N. Assume that ν = n ∈ N. When looking for solutions of (V. 1) of the form of series of ascending powers of x, one sees that J n and J -n are still solutions of (V. 1), where J n is still defined by (V. . However now J -n (x) = (-1) n J n (x), hence J n and J -n are linearly dependent, (see [44, section 3.12, p. 43] or [27, eq. (5.4.10), p. 105]). The determination of a fundamental system of solutions in this case requires further investigation. In this purpose, one introduces the Bessel's of order ν and of the second kind: among the several definitions of Bessel's functions of second order, we recall here the definition by Weber. The Bessel's functions of order ν and of second kind are denoted by Y ν and defined by (see [44, section 3.54, eq. ( 1)-( 2 For any ν ∈ R + , the two functions J ν and Y ν always are linearly independent, see [44, section 3.63, eq. ( 1), p. 76]. In particular, in the case ν = n ∈ N, the pair (J n , Y n ) forms a fundamental system of solutions of the Bessel's equation for functions of order n.

In the case ν = n ∈ N, it will be useful to expand Y n under the form of a series of ascending powers. This can be done using Hankel's formula, see [44, In the case n = 0, the first sum in (V. 4) should be set equal to zero.

V.4. Zeros of Bessel functions of order ν of the first kind. The function J ν has an infinite number of real zeros which are simple with the possible exception of x = 0 ([44, section 15.21, p. 478-479 applied to C ν = J ν ] or [27, section 5.13, Theorem 2, p. 127]). We denote by (j ν,n ) n≥1 the strictly increasing sequence of the positive zeros of J ν : 0 < j ν,1 < j ν,2 < • • • < j ν,n < . . . and we recall that j ν,n → +∞ as n → +∞. We will also use the following bounds on the zeros, proved in Lorch and Muldoon [START_REF] Lorch | Monotonic sequences related to zeros of Bessel functions[END_REF]:

(V. 5) ∀ν ∈ [0, 1 2 ], ∀n ≥ 1, π(n + ν 2 - 1 4 ) ≤ j ν,n ≤ π(n + ν 4 - 1 8
),

(V. 6) ∀ν ≥ 1 2 , ∀n ≥ 1, π(n + ν 4 - 1 8 ) ≤ j ν,n ≤ π(n + ν 2 - 1 4
).

We also mention the following inequality from [START_REF] Qu | Best possible" upper and lower bounds for the zeros of the Bessel function Jν (x)[END_REF] (V. 7) ∀ν > 0, ∀n ≥ 1, ν -a n 2 1/3 ν 1/3 < j ν,n < ν -a n 2 1/3 ν 1/3 + 3 20 a 2 n 2 1/3 ν 1/3 where a n is the n-th negative zero of the Airy function.

Moreover, we recall that it is classical ( [START_REF] Komornik | Fourier Series in Control Theory[END_REF] p. 135) that • if ν ∈ [0, 1 2 ], the sequence (j ν,n+1 -j ν,n ) n is nondecreasing and converges to π, • if ν ≥ 1 2 , the sequence (j ν,n+1 -j ν,n ) n is nonincreasing and converges to π.

V.5. Orthogonality property. For all ν ≥ 0, Bessel functions satisfy the following orthogonality property (see [27, eq. (5.14.4) and (5.14.6), p. 129]):

(V. 8)

ContentsI.

  Introduction and main results I.1. Description of the problem I.2. Main results and comments II. Well-posedness of the problem II.1. Preliminary transformation II.2. Homogeneous boundary conditions and a source term II.3. Non homogeneous boundary condition III. Proof of the main results (Theorems I.1 and I.2) III.1. Eigenvalues and eigenbasis of the Sturm-Liouville Problem III.2. Useful estimates on eigenvalues and eigenfunctions III.3. Transformation into a moment problem III.4. Formal solution of the moment problem III.

  7) IV. Technical part IV.1. Proof of Proposition III.1 IV.2. Proof of Lemma III.1 IV.3. Proof of Lemma III.2 V. Appendix : elements of Bessel Theory Date: August 15, 2017.

1 . 2 L 2

 122 Theorem 2.4 in[START_REF] Cannarsa | The cost of controlling weakly degenerate parabolic equations by boundary controls[END_REF] is formulated in the following way: (III. 14) σ + m (0,T ) ≤ C u e -2λmT e Cu √ λm γ min B(T, γ min ), with (III. 15)

1T e 32 81π 2 16 81π 2

 322162 T .Since |r mu,1 | = j ν(µ),1 ≤ π, we obtain thatC ctr-bd ≥ √ c u πT 3/2 (1 + T ) e -λµ,1T e T ,which, using the fact that λ µ,1 ≤ C(1 + ν(µ)), proves (I. 5).

  + n(µ) + 1) Γ(m + n(µ) + 1).

  the Gamma function (see[44, section 3.1, p. 40]). The first of the two series converges for all values of x and defines the so-called Bessel function of order ν and of the first kind which is denoted by J ν :(V. 2) J ν (x) := ∞ m=0 (-1) m m! Γ(m + ν + 1) x 2m+ν , x≥0, (see [44, section 3.1, (8), p. 40] or [27, eq. (5.3.2), p. 102]). The second series converges for all positive values of x and is evidently J -ν : (V. 3) J -ν (x) := ∞ m=0 (-1) m m! Γ(m -ν + 1)

  ), p. 64] or [27, eq. (5.4.5)-(5.4.6), p. 104]):    ∀ν ∈ N, Y ν (x) := J ν (x) cos(νπ) -J -ν (x) sin(νπ) , ∀n ∈ N, Y n (x) := lim ν→n Y ν (x).

  section 3.52, eq. (3), p. 62] or [27, eq. (5.5.3), p. 107]:(V. 4) ∀n ∈ N , Y n (x) + n + 1) Γ(m + n + 1) ,where Γ Γ is the logarithmic derivative of the Gamma function, and satisfies Γ (1) Γ(1) = -γ (here γ denotes Euler's constant) and Γ all m ∈ N.

1 0 1 2

 11 xJ ν (j ν,n x)J ν (j ν,m x)dx = [J ν+1 (j ν,n )] 2 if n = m, 0 if n = m.

  In that context, when there is a 'bad' upper global gap γ max , and a 'good' (much smaller) asymptotic upper gap γ * max , it is interesting to use the following extension of Theorem III.2:

1, Hence (III. 23) ∀µ ≤ 0, ∀n ≥ N * , λ µ,n+1 -λ µ,n ≤ γ * max with (III. 24) N * := [ν(µ)] + 1 and γ * max := 2π.