Alain Bossavit
email: bossavit@lgep.supelec.fr

Bulk Forces and Interface Forces in Assemblies of Magnetized Pieces of Matter

Keywords: Magnetostatics, magnetic force, contact forces, virtual power principle, Maxwell's tensor

When two magnets are stuck together, where do magnetic forces operate and which formulas should one apply to compute them? Such frequently asked questions do not find immediate answers in the literature on forces, mainly because the force field is obtained, by the Virtual Power Principle (VPP), as a (mathematical, vector-valued) distribution, not as a plain vector field, which would be more convenient for practical computation. We intend to show, in a few important cases of contact (between two linear materials with different permeabilities, between magnet and magnetizable metal, linear or not, between two hard magnets, etc.), how to represent this single distribution by two vector fields, one of them borne by the bulk of the matter, the other one localized at material interfaces where discontinuities of permeability, of magnetization, etc., do occur. Suitable extensions of the classical Maxwell tensor play an important role in the computation (by the so-called 'pillbox trick') of this interface vector field.

I. INTRODUCTION

C ONSIDER a piece of matter with reluctivity ν (indepen- dent on the magnetic field) plunged into the field of a DC coil that lies some distance away. 1 Suppose ν insensitive also to the local strain (this is to avoid for the moment the difficulties of magnetostriction), but possibly non-uniform inside the domain D occupied by the matter. One will learn, from various sources (e.g., [START_REF] Maugin | The principle of virtual power in continuum mechanics: Application to coupled fields[END_REF], [START_REF] Robinson | Macroscopic Electromagnetism[END_REF], [START_REF] Carter | Distribution of mechanical forces in magnetised material[END_REF], etc.), that the force field inside D, or 'bulk force', is 1 / 2 |B| 2 ∇ν. This vanishes for uniform ν. Yet the piece is attracted by the coil, so there must be a force, which cannot reside elsewhere than at the air-matter interface S, the boundary of D (Fig. 1). Indeed, it can be shown (we do it in detail below) that this surface force is

F S = 1 / 2 (|H τ | 2 [µ] -|B n | 2 [ν]) n, (1)
where n is the outward unit normal, H τ and B n the tangential part of H and normal part of B (both continuous across S), and [µ] and [ν] the jumps of µ and ν across the surface. (Note that [µ] > 0, as a rule, and hence [ν] < 0. Look at Fig. 1 for the sign conventions about the jump.) There are several ways to prove [START_REF] Maugin | The principle of virtual power in continuum mechanics: Application to coupled fields[END_REF]. One of them consists in taking 1 / 2 |B| 2 ∇ν 'in the sense of distributions'. I shall explain in detail what this means, but let us first see how the Virtual Power Principle (VPP) yields the magnetic force as a distribution, by its very nature.

Let the symbol v (letter 'vee', not to be confused with ν for 'nu') denote the velocity of a virtual motion, in which a particle sitting at point x in the reference configuration (the one for which we want to compute forces) is displaced to the point x + tv(x) at time t. We take v smooth and compactly supported (i.e., null outside some bounded region, called the support of v). Call Ψ v (t, B) the magnetic energy the system would contain at time t of this virtual motion, if the magnetic induction was B at this instant. Then-a well-known result; cf. [START_REF] Bossavit | On forces in magnetized matter[END_REF] for a detailed proof-the virtual power at time 0 is minus the partial derivative of Ψ v with respect to t, for t = 0 and B = B(0), its value at time 0. Hence a linear function of v. It may happen that this function (which yields the virtual power at t = 0 for the virtual motion associated with v) has the form F • v, where F is a vector field, which is then, by definition, the force field. (All integrals of this kind, where the integration domain is left unspecified, are over all space.) But most often the map v → -∂ t Ψ v (0, B(0) is just that: a map, linear in v, with the kind of continuity with respect to v that qualifies it as a distribution. (It's a vector-valued distribution, since the test functions v are themselves vector-valued.)

For instance, in the case just evoked of a piece with reluctivity ν, the linear map one finds cannot be written as F • v, where F would be 1 / 2 |B| 2 ∇ν at all points where this vector field is well defined. This would exclude S, across which both ν and |B| 2 are discontinuous, and thus would make us miss the surface force. The convenient notation 1 / 2 |B| 2 ∇ν will be used nonetheless for the force distribution, but it will denote a different object than F . Which object, exactly, is what we need to make clear, and the proof of (1) will come as a by-product. This exercise will be followed by a more difficult one, the case of hard magnets with B = µ 0 (H + M ) as B-H law. Several possibilities exist for how M depends on the deformation of matter. One of them (the simplest, according to which M rotates with matter, but does not depend on local strain) was addressed in [START_REF] Bossavit | On forces in magnetized matter[END_REF], where the force field 'in the sense of distributions' was found to be

F = -∇M • B -1 / 2 rot(H × B), (2)
where the meaning of ∇M •B will soon be made precise. There is again, hidden in (2), a system of forces borne by S (the air-magnet interface, or the magnet-magnet interface where M can be discontinuous), a part of which is normal to S and the other part tangential. We shall generalize all that to a large category of non-linear B-H laws. On notations: Given vector fields X and Y , we define X•∇Y as the vector field with components X j ∂ j Y i in an orthonormal frame, using Einstein's convention. In contrast, ∇Y • X is

∂ i Y j X j , as in (2). One will check that ∇Y • X-X • ∇Y = X × rot Y. The 'dyadic product' X ⊗ Y is the 2-tensor T with components T ij = X i Y j .
Other examples of 2-tensors are the above ∇Y and the Kronecker δ. The classical Maxwell tensor is then M T = H ⊗ B -1 / 2 (H •B) δ. We shall have use for the right-side divergence div T , in components ∂ j T ij , of a 2-tensor T , and for its product T • X, in components T ij X j , with a vector X. Note that div δ = 0 and div(f δ) = ∇f if f is a scalar field.

II. DISTRIBUTIONS 101

What follows is an elementary introduction to aspects of distribution theory relevant here. Let us for a moment distance ourselves from electromagnetism and deal with two scalar functions f and g. (Later, they will become ν and |B| 2 .) The test functions, smooth and compactly supported, are called ϕ when scalar-valued, v when vector-valued.

If f is just integrable, without more regularity, the map ϕ → f ϕ does qualify as a distribution. But for a distribution such as the 'Dirac mass' ϕ → ϕ(a), where a is a given spatial point, there is no function δ a such that ϕ(a) = δ a ϕ. Thus, distributions encompass functions and generalize them [START_REF] Schwartz | Théorie des distributions[END_REF].

When f is not differentiable, its gradient 'in the sense of distributions' exists nonetheless. It's the distribution v → -f div v, to which one extends the notation ∇f . Then one understands ∇f • v as -f div v. A notational abuse, of course, but which makes sense: if f were differentiable all over space, one would have -f div v = ∇f • v, indeed. Now suppose f smooth inside both D and the outer region D , but discontinuous across their common boundary S, with a jump [f ]. Then, integrating by parts on D and D ,

∇f •v =-f div v = - S [f ] n•v+ D∪D (∇ s f )•v, (3)
where ∇ s f denotes the 'strong' gradient of f , well-defined in D and D , but not on S. The capped equal sign means that the integral expression to its left is defined as the one to its right.

So the vector-valued distribution denoted by ∇f in (3) can be represented by two ordinary vector fields, the almost everywhere defined ∇ s f , living on 3D space, and [f ] n, living on S only. We find them, as a rule, in the roles of body force and interface force, in all situations evoked here. Now, let's try and interpret in the sense of distributions the product g ∇f . If both f and g are smooth all over and compactly supported, one has g ∇f • v = -f div(gv). So we may take that as a definition of g ∇f, provided f div(gv) makes sense, which requires [g] = 0. Then,

(g ∇f ) • v = - S g [f ] n • v + D∪D g (∇ s f ) • v. (4) D S n S x B(x) B (x) τ α D' B (x) n n Fig. 1.
The jump [g] of a scalar quantity g across S is its value on the 'upstream' side of S minus its value on the 'downstream' side, as both defined by the direction of the normal field n. By convention, n goes from D to D here. Also shown, one of the surfaces Sα of the foliation described in the text (S is S 0 ) and suggested, the orthogonal decomposition of the field B into normal and tangential parts.

The constraint [g] = 0 is not a surprise, since the product of two distributions (here g and ∇f ) does not exist unconditionally. But our goal, to find an interpretation of |B| 2 ∇ν as a distribution, is thwarted, since [|B| 2 ] = 0 as a rule. Neither can we handle -|H| 2 ∇µ that way, since [|H| 2 ] = 0 as well. Fortunately, a suitable combination of these two expressions will work.

III. PROVING (1)

Suppose the interface S presented as the locus of points x for which s(x) = 0, for some smooth real function s. (Having that locally is enough.) Then, the surfaces S α = {x : s(x) = α}, for α in a neighborhood of zero, say -δ < α < δ, make a foliation of a neighborhood of S, call it D δ . Call d the function on D δ defined by d(x) = α when x belongs to S α . To each such point x, assign the unit vector (∇d)(x)/|(∇d)(x)|, hence a field n which extends the field of unit normals to S considered so far. Pick also two unit vectors, anchored at x, tangent to S d(x) , mutually orthogonal, both smoothly depending on x. This way, we have a smooth system of orthonormal frames, 'adapted' to S in an obvious sense. Any smooth vector field X will have (when restricted to D δ ) an orthogonal decomposition of the form X = X n n + X τ , normal part plus tangential part. When X is smooth in D and D separately, but discontinuous across S, one may talk of the jumps [X n ] n and [X τ ] across S of these two parts. Now, let's apply this to the physical fields H and B, for which [H τ ] = 0 and [B n ] = 0. Over D δ , except on S where discontinuities occur, we have

|B| 2 ∇ν = B 2 n ∇ν + |B τ | 2 ∇ν = B 2 n ∇ν + |µH τ | 2 ∇ν = B 2 n ∇ν + |H τ | 2 µ 2 ∇ν = B 2 n ∇ν -|H τ | 2 ∇µ, (5)
since µ∇ν = -ν∇µ as entailed by νµ = 1. Thus, |B| 2 ∇ν appears as the difference of two terms, B 2 n ∇ν and |H τ | 2 ∇µ, both of the form g∇f with [g] = 0 on which we worked previously. Applying (4) to both terms yields [START_REF] Maugin | The principle of virtual power in continuum mechanics: Application to coupled fields[END_REF].

S Σ n n Σ S ν 2 ν 1 [ν] = ν -ν 2 1
Fig. 2. Surface Σ here is astride a part of the material interface S. Integrating T M n over Σ gives the magnetic force on this part. The top and bottom surfaces of the pillbox need not coincide with surfaces of the foliation Sα of Fig. 1 (but should be close to S on the side of ν i in case ν i , i = 1, 2, is not uniform).

IV. MAXWELL'S TENSOR AND THE 'PILLBOX' METHOD

There is another approach to the previous result, by using the identity, valid when H = νB and divB = 0,

div( M T ) = (rotH) × B + 1 / 2 |B| 2 ∇ν. (6)
This is just algebra when ν, H and B are smooth, and we allow discontinuities by taking [START_REF] Sánchez Grandía | General Formulation For Magnetic Forces in Linear Materials and Permanent Magnets[END_REF] in the sense of distributions. Consider now the vector field M T • n Σ , i.e., ( M T ) ij n j Σ in coordinates, where n Σ is the normal field living on the surface Σ of Fig. 2. Integrate it over the pillbox surface Σ. The result is a vector. According to Gauss's theorem and our previously acquired knowledge (cf., e.g., [START_REF] Bossavit | On forces in magnetized matter[END_REF]) that (rotH)×B+ 1 / 2 |B| 2 ∇ν is the bulk force, this vector is the total force upon the matter inside Σ, and it converges towards the force on the part of the interface within Σ when the top and bottom surfaces of the pillbox are moved towards S. (Piecewise smoothness is enough for proving this convergence.)

The passage to the limit thus suggested leads to (1) by a computation one may find more intuitive than the previous one. The reason for that is visible in (6): As a distribution, M T is more regular (it's an ordinary tensor field, actually, almost everywhere defined) than its divergence (rotH) × B + 1 / 2 |B| 2 ∇ν. This is an example of the general and well known fact that integration has a smoothing effect, contrary to differentiation. (Think of the vector potential A, more regular than the field B that derives from it: Here, M T is a 'tensor potential' from which the right-hand side of (6) derives. The analogy is apt in many respects.)

It thus appears that the knowledge of a 2-tensor from which the force, as a distribution, derives, is desirable. Hence our working programme from this point on: In more and more complex situations (non-linearity of the B-H law, anisotropy, magnetostrictive behavior), find the force F as a distribution by applying the VPP, then find a 2-tensor T such that div T = F (clearly, T need not be unique), then integrate T • n Σ over the surface of a suitable pillbox to reveal the interface forces.

V. MAGNETIC ENERGY DURING A VIRTUAL MOTION

Let us assume that we face a problem in magnetoelasticity in which the magnetic energy density of a piece of matter depends on its state of deformation β and of the ambient magnetic field b. (This excludes magnetic hysteresis and a lot of complex mechanical behaviors, such as those of ferrofluids and other so-called polar materials.) So we have a function ψ(x, β, b) where β is a linear map (the one that sends a material vector ξ anchored at point x to its new value β(ξ) in the deformed state), and b a vector. This function, we assume, is convex with respect to b, and the B-H law is simply expressed by H(x) = ∂ b ψ(x, β(x), B(x)). The magnetic energy density (later denoted by ψ) is then ψ(x) = ψ(x, β(x), B(x))dx, and the total magnetic energy is ∫ ψ(x)dx, an integral over all space, with dx as volume element.

Note that β can be considered as a 2-tensor, but not a symmetric one. By polar decomposition, one can write β = rs, where r is a rotation and s a symmetric map, the one known as strain in Mechanics. As a very general rule, constitutive laws are not affected by rotations, which in our case implies that ψ(x, rs, rb) = ψ(x, s, b) whatever the rotation r. It is therefore enough to know ψ(x, s, b), where s is a symmetric 2-tensor. We shall have use for the three partial derivatives of ψ, also as functions of x, s and b, denoted by ∂ x ψ, ∂ s ψ, ∂ b ψ. (Occasionally, we may write ∇ψ instead of ∂ x ψ, since ∇ connotes spatial derivation usually.) We note that ∂ s ψ has the dimension of a stress, which justifies calling σ M , with M for 'magnetostriction', its value in the reference state.

During a virtual motion x → x+tv(x), the magnetic energy changes, even though B is frozen at its reference value, because the material configuration changes. So we call ψ v (t, x, B(x)) the value of the magnetic energy density at time t and point x in the ambient field B(x) during such a virtual motion. (No mention of β: it is determined by v. Beware that ψ and ψ v are very different objects.) Recall (cf. [START_REF] Bossavit | On forces in magnetized matter[END_REF]) that the virtual power we aim at is the partial derivative in t, at t = 0, of (minus) the total magnetic energy Ψ v (t, B) = ∫ ψ v (t, x, B(x))dx, with B frozen in all space at its value at t = 0.

We start from

ψ v (t, x + tv(x), b) = ψ(x, s v (t, x), [r v (t, x)] -1 b), (7)
where s v and r v are the elements of the deformation (strain and rotation) all over. What [START_REF] Schwartz | Théorie des distributions[END_REF] says is this: "The particle that was at point x at time 0 has reached the point x + tv(x) at time t, and because of the deformation r v (t)s v (t) that it has sustained, the magnetic energy density it bears in field b is what it would have been at x under the strain s v in the field [r v (t, x)] -1 b." Next, we obtain the time-derivative of -ψ v at time 0 by differentiating both sides of (7) in t, using the chain rule. (Hence the virtual power, by summing up -∂ t=0 ψ v over all space.) One makes use of the approximations s v (t, x) 1+t∇ sym v and r v (t, x)b 1+t/2 (rotv)×b, where ∇ sym v is the symmetrized 2-tensor (∂ i v j + ∂ j v i )/2. This amounts to neglecting terms in t 2 and higher order in the Taylor expansion of -ψ v about t = 0, which doesn't affect the result:

-∂ t ψ v | t=0 = v • [∇ψ + div(σ M ) -1 / 2 rot(H × B)], (8)
as obtained when one substitutes B(x) for b. The appearance of H there is due to the fact that H = ∂ b ψ. The operators rot and -div come from standard integration by parts, as adjuncts of the rot in "rot v" and of the ∇ sym in "∇ sym v".

Remarkably, the result is generic: Whatever the B-H law, one has the same three terms in the expression of the magnetic force (in addition to the (rotH) × B force): An inhomogeneity term ∇ψ (generalizing the |B| 2 /2 ∇ν we had when ψ was ψ(x, s, b) = 1 / 2 ν(x)|b| 2 whatever s), a magnetostrictive term div(σ M ), and an anisotropy term -1 / 2 rot(H × B). All of these, of course, to be interpreted as distributions, so we aim now at a Maxwell-like tensor 2 whose divergence would be the term between square brackets in [START_REF] Bermúdez | Extended Formulas to Compute Global and Contact Electromagnetic Force and Torque from Maxwell Stress Tensors: The Role of Symmetry[END_REF].

VI. AN AUGMENTED MAXWELL TENSOR

Let's recall a few identities: For H and B not necessarily the physical fields by this name,

(rot H) × B = B • ∇H -∇H • B. (9)
One remarks that

div(H ⊗ B) = H divB + B • ∇H, (10)
div[H ⊗ B -B ⊗ H] = rot(H × B). (11)
Let now B and H be the physical fields, linked by the law ψ(B(x)) + ϕ(H(x)) = B(x) • H(x), where ϕ is the coenergy density dual to ψ, with div B = 0. (We'll say that B and H make a 'magnetic pair'.) To avoid confusion, we denote the ∇ψ of [START_REF] Bermúdez | Extended Formulas to Compute Global and Contact Electromagnetic Force and Torque from Maxwell Stress Tensors: The Role of Symmetry[END_REF], that is to say the partial derivative of ψ with respect to position in presence of the ambient field B, by ∇ψ(•, B). Same convention for ∇ϕ(•, H). Recall that ∇ψ(•, B) = -∇ϕ(•, H) for such a magnetic pair. Let us denote by ϕ(H) the coenergy density ϕ(x, H(x)), a function of x only (the H is just a label). By the chain rule,

∇ ϕ(H) = ∇ϕ( • , H) + ∇H • B. (12)
We may now assert:

Proposition 1. If B and H form a magnetic pair, then

div[ 1 / 2 (H ⊗ B + B ⊗ H) -ϕ(H) δ] = (13)
(rot H) × B -∇ϕ( • , H) -1 / 2 rot(H × B).
Proof of (13): Thanks to (9), (rot

H)×B-∇ϕ( • , H) = B•∇H -(∇ϕ( • , H) + ∇H • B) = B • ∇H -∇ ϕ(H), taking (12
) into account. But this is div(H ⊗ B -ϕ(H) δ), thanks to (10) and to the fact that div(g δ) = ∇g. So one has

div(H ⊗ B -ϕ(H) δ) = (rot H) × B -∇ϕ( • , H). ( 14
)
Subtracting from (14), on both sides, half of (11), one does find (13). Hence a Maxwell tensor that generalizes Sanchez et al.'s proposal in [START_REF] Sanchez-Grandia | Magnetostatic Maxwell's tensors in magnetic media applying virtual works method from either energy or co-energy[END_REF] and [START_REF] Sánchez Grandía | General Formulation For Magnetic Forces in Linear Materials and Permanent Magnets[END_REF] (which was H ⊗ B -ϕ(H) δ, if expressed in our notation) by accounting for the anisotropy forces (whose omission may make observable difference, as shown in [START_REF] Bermúdez | Extended Formulas to Compute Global and Contact Electromagnetic Force and Torque from Maxwell Stress Tensors: The Role of Symmetry[END_REF]). By adding σ M to it, we obtain the following augmented Maxwell tensor,

A T = 1 / 2 (H ⊗ B + B ⊗ H) -ϕ(H) δ + σ M , (15)
which also accounts for magnetostriction. 2 For lack of space, Section VI that follows had to be excised from the published version, in which the expression (15) for the augmented Maxwell tensor was given without proof.

VII. INTERFACE TERMS

Let us conclude by giving the results of some pillbox computations. In the nonlinear but isotropic case, we find F S = ([ ϕ(H)] -[H n ]B n )n by using the adapted system of frames of Fig. 1 ([ ] denotes the jump). There is an equivalent 'Boriented' formula, F S = (-[ ψ(B)] + [B τ ] • H τ )n, showing that there is no "preference for the coenergy" in the whole approach. (Details on this will appear in [START_REF] Bossavit | Un tenseur de Maxwell non linéaire et symétrique[END_REF].)

In case of anisotropy, the term -1 / 2 rot(H × B) in (2) stands for a distribution represented by two fields. One is -1 / 2 rot s (H × B), with the strong form of the curl, the bulk force. The other one, borne by S, is expressed by one half the jump [n × (H × B)]. By the double cross product formula, this jump is

[(n • B)H -(n • H)B], which equals n × [n • H n × B].
This can be seen to reduce to -1 / 2 [B τ H n ], a tangential field living on S, by using the S-adapted system of frames.

Especially interesting is the case B = µ 0 (H + M ). In the adapted frame system, M = M n n + M τ , both M n and M τ smooth in D and D , but discontinuous, with jumps [M n ] and [M τ ] across S. Substituting ν 0 B -M for H and µ 0 (H + M ) for B, one finds, after a short calculation,

[n × (H × B)] = µ 0 [M n ]H τ -B n [M τ ] + µ 0 [M n M τ ], (16)
to be multiplied by 1/2 to get the tangential part of the interface force F S . Last, the term -∇M • B of (2) is found, by the technique of Section II, to contribute to F S the normal field

{[M n ]B n + µ 0 [M τ ] • H τ + 1 / 2 µ 0 [|M τ | 2 ]}n. (17)
It would remain to explore the large realm of magnetostriction by using (15). For example, assume B = µ 0 (H + M (s)). Then σ M = -∂ s (M (s) • B). One finds the same surface forces as above, with M = M (1), plus a term in [σ M • n]. There is also a bulk force div(σ M ), even if M is uniform (because B is not, as a rule). Knowing only M (1), the magnetization in the reference configuration, would not give access to these terms, thus leaving the question "What is the force inside a permanent magnet?" ill-posed.

To sum up: Interface forces can be obtained by the VPP, but as distributions, and these are a delicate tool. More robust (and fully equivalent) is the use of Maxwell-like tensors allied with the pillbox trick. But such tensors, built from the VPPcomputed forces, not the other way round, may differ from the classical Maxwell tensor, which should not be construed as a primary concept in the question of forces, but as just a helpful analytical tool.

Compumag

paper, submitted to IEEE Transactions on Magnetism, accepted July 1, 2015. Date of current version: Jan. 4,

Apart from cosmetic edits, the main difference with the published one is the reinsertion of Section VI, formerly cut off for lack of space.


Stanislaw Gorlow
email: stanislaw.gorlow@gmail.com

Coding Backward Compatible Audio Objects with Predictable Quality in a Very Spatial Way

A gradual transition from channel-based to object-based audio can currently be observed throughout the film and the broadcast industries. One paramount example of this trend is the new MPEG-H 3D Audio standard, which is under development. Other object-based standards in the market place are DTS:X and Dolby Atmos. In this engineering brief, a newly developed prototype of an object-based audio coding system is introduced and discussed in terms of its technical characteristics. The codec can be of use everywhere where a given sound scene is to be re-rendered according to the listener's preference or environment in a backward compatible manner. The areas of application cover not only interactive music listening or remixing, but also location-dependent, immersive, and 3D audio rendering.

Introduction

Object-based audio coding is a more recent field of research that poses a challenge to the existing audio coding approaches. Its main objective is to represent a complex sound scene consisting of many different sound sources in a compact and reversible manner. A certain degree of quality loss is usually tolerated [START_REF] Gorlow | Informed Audio Source Separation Using Linearly Constrained Spatial Filters[END_REF].

Not so long ago, a coding system for multitrack music recordings was presented in [START_REF] Marchand | DReaM: A Novel System for Joint Source Separation and Multi-Track Coding[END_REF]. The output format in the case of music is typically stereo. This means that the mixture signal space is limited to two dimensions. As a consequence, the quality of the separated tracks has an upper bound that is determined by the spectral overlap and 2 nd -order dynamics of the source signals, the spatial distribution of the sound sources, and their presence in the mixture in terms of loudness. In [START_REF] Gorlow | On the Informed Source Separation Approach for Interactive Remixing in Stereo[END_REF] it was shown that the system was sensitive to perceptual audio coding, such as MP3 or AAC. The two-channel mixing approach was extended to an arbitrary number of channels in [START_REF] Gorlow | Multichannel Object-Based Audio Coding with Controllable Quality[END_REF]. There, it was shown that and how a set of single-channel source signals can be combined to form a multi-channel mixture and then separated or unmixed with the signal quality at the output being set in advance, i.e. prior to (down-)mixing. Accordingly, the compression ratio is given as the ratio between the number of sources and the number of channels.

In this brief, a revised coding system is presented that remedies the shortcomings of [START_REF] Marchand | DReaM: A Novel System for Joint Source Separation and Multi-Track Coding[END_REF]. These are:

1. The intrinsic quality limitation that is due to the fixed number of mixture channels;

2. The audible degradation of sound quality that is due to perceptual audio coding.

In addition, the findings from [START_REF] Gorlow | Informed Separation of Spatial Images of Stereo Music Recordings Using Second-Order Statistics[END_REF] were also taken into account during the design of the system. As an extra, dynamic range compression/decompression [START_REF] Gorlow | Model-Based Inversion of Dynamic Range Compression[END_REF] can be integrated in the new codec on demand [START_REF] Gorlow | Reverse Engineering Stereo Music Recordings Pursuing an Informed Two-Stage Approach[END_REF].

System Overview

In this section, the overall coding system is presented in the form of block diagrams (encoder and decoder).

Encoder

The encoder is depicted in Fig. 1.

Object Analysis

Decoder

The decoder is depicted in Fig. 2.

System Properties

In this section, the most significant properties of the new coding system are elaborated in more detail. These are:

Flexible object format The audio objects can be single-channel or multi-channel. The static background of an acoustic scene, the sound canvas, would typically be represented as a multi-channel object, while the (active) objects (on top) would typically be single-channel.

Perception-aware quality metric The metric used for quality control is both spatial and perceptual. And so it gives a good estimate of the sound quality.

Predictable sound quality With the help of the quality metric it is possible to predict the attainable sound quality after decoding prior to downmixing.

Flexible mode of operation

The codec can be operated in three different modes:

1. Regular mode. In this mode, only the artistic mixture is generated from the canvas and the objects. This is for (passive) listening.

2. Virtual mode. In this mode, a synthetic mix is generated, which maximizes the quality with a minimum number of virtual channels. This is for data reduction, but not for listening.

3. Hybrid mode. This is the default, or where the magic happens. This mode combines the two previous modes: the regular mixture channels are increased by the virtual channels in such a way that the desired quality is attained with a minimum number of virtual channels. This is the mode that enables both passive and active listening in high quality.

Coding efficiency The (spatial) coding is typically such that the total number of channels is at most equal to the number of single-channel objects. Accordingly, a stereo canvas would be counted as two objects. As an option, the number of virtual channels can be set to any number or determined a priori by the quality control mechanism and held fixed during the entire encoding process, which leads to a constant channel rate (CCR), or their number can be determined on a block-by-block basis and varied, thus resulting in a variable channel rate (VCR). This approach allows for the lowest average channel rate (ACR) w.r.t. the desired quality, as the number of channels and thus the spatial resolution is adapted locally to the signal constellation, which, in other words, translates into the number of active objects, the spectral dynamics, the spatial locations, and the relative volume levels. And since the content from the regular mix is used as well, the amount of virtual data is much smaller than what it would be, if all the objects were coded separately. To reduce the data rate even further, the (spatial) coder can be used in tandem with any type of (perceptual) audio coder. In that case, it acts as a spatial pre-processor for the encoder and as a post-processor for the respective decoder, accordingly. 4Digital rights management The regular and the virtual channels alike, both carry realizations of mixtures, which are generated from the audio objects.

In the case of the virtual channels, these realizations may even be "sparse". This accounts for the reduction of the overall data rate. This can be seen as some sort of implicit copy protection, because the objects are separately inaccessible without the decoder. Access can further be restricted in such a way that distinct objects cannot be soloed during playback, etc.

Audio objects over IP The spatial data that enables the system to meet the quality constraint and the meta-data are generated on a frame-by-frame basis. This is also the case for most if not all trans-form coders, such as MP3. Thus, the codec can also be used for audio streaming.

Mixing and mastering

The mixing stage in the encoder may be followed by a mastering stage. This applies only to the artistic mixture. The mastering stage, in a typical scenario, would comprise a multi-band or single-band compressor and a limiter, possibly with look-ahead. These elements would then be undone on the decoder side through inverse modeling. This allows for the creation of mixes that are even more realistic and so, closer to the CD or the radio edited version of the same multi-track.

Possible Applications

In this section, some possible applications for the codec are mentioned. They mainly relate to:

Music publishing Consider the case where the mixing engineer exports the tracks (bass, guitar, vocal, etc.) that are to be mixed together and notes down the panning and the gains of the tracks. The mastering engineer does the mastering on a premixed version of the multi-track and notes down the compressor model and the respective parameters. Then, to generate a re-mixable version of the song, the encoder, which could also be a plugin for the DAW, would take the tracks as input together with the mixing and the mastering parameters. It would generate all the necessary side information and eventually generate the downmix. On the consumer side, the corresponding decoder would take the downmix and the side information as input. The side information would grant access to the composite tracks by separating them from each other in the mix. A re-rendering interface would allow the user to manipulate an apparently rendered piece of music without the direct access to the separate tracks. A concrete example is a vehicle with a 3D sound system. Imagine you buy a CD of your favorite artist that was "enriched" by the technology. You could choose to either listen to the stereo sound or you could switch on the AR mode and have the sound in 3D with the option to manipulate the spatial sound according to your personal taste.

Other applications would typically focus on new ways of delivering user-centered audio content with the aim of a greater audience engagement.

Summary

In this brief, a new multichannel object-based audio coding scheme was presented. Its key features are:

• Backward compatibility,

• Scalable data size and sound quality,

• Robustness against perceptual audio coding.

Its possible applications lie in the area of digital music publishing and 3D sound rendering related to AR/VR. More applications should emerge in the future.

Fig. 1 :

 1 Fig. 1: Multichannel object-based audio encoder

1Fig. 2 :

 2 Fig. 2: Multichannel object-based audio decoder

Since the virtual channels contain (coded) audio, it might even be possible to use DTS' Multi-Dimensional Audio (MDA) open format.

Acknowledgments

This work was partially funded by "Aquitaine Science Transfert" through the Caumbot project (PJ 2014-089).


Pierre Lissy
email: lissy@ceremade.jussieu.fr

Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory

In this paper, we construct some interesting Gevrey functions of order α for every α > 1 with compact support by a clever use of the Bray-Mandelbrojt iterative process. We then apply these results to the moment method, which will enable us to derive some upper bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type that partially improve the existing results proved in [P. Lissy, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension SIAM J. Control Optim., 52(4), . However this construction fails to improve the results of [G. Tenenbaum and M. Tucsnak, New blow-up rates of fast controls for the Schrödinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100] in the precise case of the usual heat and Schrödinger equation.

Introduction

Presentation

The main motivation of this paper is to continue the study of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] concerning the estimation for the cost of fast "boundary" controls for a class of linear equations of parabolic or dispersive type. The precise scope of the paper will be made more precise later.

Let us introduce some usual notations. Let H be some Hilbert space and U be another Hilbert space. Let A : D(A) → H be a self-adjoint operator with compact resolvent. The eigenvalues of A (which are here supposed to be different from 0 without loss of generality) are called (λ k ) k 1 and supposed to be with multiplicity 1 in all what follows. To each eigenvalue we associate a normalized eigenvector corresponding to the eigenvalue λ k , which is called e k . We assume that -A generates on H a strongly continuous semigroup S : t → S(t) = e -tA . The Hilbert space D(A * ) (= D(A) ) will be equipped with the norm

||x|| 2 D(A) = < x, e k > 2 H λ 2 k .
We call B ∈ L c (U, D(A) ) an admissible control operator for this semigroup, i.e. such that for every time T > 0, there exists some constant C(T ) > 0 such that for every z ∈ D(A), one has

T 0 ||B * S(t) * z|| 2 U C(T )||z|| 2 H .
We consider the following class of controlled semigroups:

y t + Ay = Bu (parabolic case) (1)
and

y t + iAy = Bu (dispersive case). (2)
In the case of Equation ( 1), A will moreover be supposed to be a positive operator to ensure the existence of solutions. Then, it is well-known (see for example [2, Chapter 2, Section 2.3], the operators -A or -iA generates a strongly continuous semigroup under the hypothesis given before thanks to the Lummer-Phillips or Stone theorems) that if u ∈ L 2 ((0, T ), U ), System (1) or [START_REF] Coron | Control and nonlinearity[END_REF] with initial condition y 0 ∈ H has a unique solution satisfying y ∈ C 0 ([0, T ], H).

From now on and until the end of the paper, we will assume that B is a scalar control, that is to say of the form Bu = bu, where b ∈ D(A) and u ∈ L 2 ((0, T ), K) where here U = K with K = R or C. From now on, we will call

b k =< b, e k > (D(A) ,D(A)) ,
where <, > (D(A) ,D(A)) is here the duality product between D(A) and D(A) with pivot space H. It is well-known (see [START_REF] Ho | Admissible input elements for systems in Hilbert space and a Carleson measure criterion[END_REF]

) that if ||(b k ) k∈N * || ∞ < +∞ and if (λ k ) k 1 is regular in the sense that inf m =n |λ m -λ n | > 0,
then B is an admissible control operator.

Let us now introduce the notion of cost of the control. Assume that that system (1) or ( 2) is null controllable at some time T 0 > 0 (i.e. for every y 0 ∈ H, there exists some control u ∈ L 2 ((0, T 0 ), U ) such that y(T 0 , •) ≡ 0). One can then verify easily that there exists a unique control u opt ∈ L 2 ((0, T 0 ), U ) with minimal L 2 ((0, T 0 ), U )-norm. Moreover, the map y 0 → u opt is linear continuous (see for example [2, Chapter 2, Section 2.3]). The norm of this operator will be from now on called the cost of the control at time T 0 and denoted C T0 in this section. Thanks to the definition of C T0 , this constant is also exactly the smallest constant C > 0 such that for every y 0 ∈ H, there exists some control u driving y 0 to 0 at time T 0 with ||u|| L 2 ((0,T0),U ) C||y 0 || H .

Here we attend to give some precise upper bounds on C T when T → 0 for some families of operators A for which equations ( 1) and ( 2) are null controllable in arbitrary small time. Some applications to fractional heat and Schrödinger equations will be provided later.

Let us explain now precisely the scope of the paper:

1. We give a new family of possible multipliers for the moment method that depend on the asymptotic growth of (λ k ) k 1 and seem well-adapted (this is the main result of the paper).

The author believes that it would be quite difficult to improve the construction given here, notably because the construction approximates quite well some optimal problem (see Section 3.2), but there is still some small possibility of doing better (see Remark 4) that the author did not manage to exploit suitably.

2. We manage to extend the results of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] to the wider possible range of exponents for the asymptotic polynomial behavior of the eigenvalues (λ k ) k∈N * . In section 2.3, we will explain into more details why the multiplier used in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] was not really adapted to the case of eigenvalues λ k behaving roughly like k α with α = 2 and how we managed to improve it.

3. As we will see later (see notably Figures 1 and3), we will not improve the upper bounds in the case of the classical heat equation and Schrödinger equation given in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]. However, we will also see that there are a wide range of families of eigenvalues for which our estimates are better than in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], and notably when α → ∞ we dramatically improve the results of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] and we are quite close to the lower bounds of [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 transport-diffusion equation[END_REF].

State of the art

As far as the author know, the study of the behavior of fast control for partial differential equations started with the paper [START_REF] Seidman | Two results on exact boundary control of parabolic equations[END_REF], where an upper bound for the cost of fast boundary control for the one-dimensional heat equation was given. A lower bound were then given in [START_REF] Güichal | A lower bound of the norm of the control operator for the heat equation[END_REF], proving that the cost of fast controls had to be necessarily roughly (up to fractional terms in T ) of the form exp(K/T ) for some K > 0 as T → 0. The same result for the boundary control of onedimensional Schrödinger equations was proved later in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF]. The next natural question is then to try to estimate precisely K and notably its dependence with respect to the geometry of the problem (i.e. to the behavior on the eigenvalues λ k ). In the case of the heat equation, if we call L the length of the interval on which we control our one-dimensional equation, we know that

L 2 /2 K 3L 2 /4.
The best upper bound was obtained in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] whereas the lower bound was obtained in [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 transport-diffusion equation[END_REF]. For the Schrödinger equation, we have

L 2 /4 K 3L 2 /2.
The upper bound was obtained in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] and the lower bound in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF].

Let us now explain with more details what was precisely done [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]. In this article, the author proved precise upper bounds concerning the cost of the control for some large classes of linear parabolic or dispersive equations when the time T goes to 0, where the underlying "elliptic" operator was chosen to be self-adjoint or skew-adjoint with eigenvalues roughly as Rk α or ±Rk α (only for (2)) for some R > 0 and α 2 when k → +∞. The cost of the control is proved to be bounded from above by exp K/(RT ) 1/(α-1) where K is some explicit constant depending on α. This does not cover all possible cases, because it is well-known that equations like (1) and ( 2) are controllable in arbitrary small time if and only if α > 1.

Explicit lower bounds have later been derived in [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 transport-diffusion equation[END_REF] for all controllable in small time fractional heat or Schrödinger equations. Let us mention that in the case of the Schrödinger equation (which corresponds to λ k = k 2 and equation ( 2)), the lower bound proved in [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 transport-diffusion equation[END_REF] is exactly the same as the one given by Miller in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF], whereas the lower bound for the heat equation controlled on the boundary (which corresponds to the case λ k = k 2 and equation [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]) is twice the one obtained by Miller in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] and was conjectured to be to exact behavior for the cost of the control until now.

To finish, let us mention that the study of the behavior of fast controls may also be applied in some cases to study the uniform controllability of convection-diffusion equations in the vanishing viscosity limit as explained in [START_REF] Lissy | A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation[END_REF] and [START_REF] Lissy | An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit[END_REF], however the results given in the present article does not enable us to deduce directly new results for this problem.

Main results and comments

In this section, we are going to give the main results of this paper and some additional comments. Let us first set some notations. In all what follows, f g (with f and g some complex valued functions depending on some variable x in some set S) means that there exists some constant C > 0 such that for all x ∈ S, one has |f (x)| C|g(x)|, (such a C is called an implicit constant in the inequality f g), and f g means that we have both f g and g f . Sometimes, when it is needed, we might detail the dependence of the implicit constant with respect to some parameters. Theorem 2.1 Assume that (λ n ) n 1 is a regular increasing sequence of positive numbers verifying moreover that there exist some α > 1 and some R > 0 such that

λ n = Rn α + O n→∞ (n α-1 ) (3)
holds (as n → ∞), and assume that b n 1 (in the sense that the sequence (|b n |) n∈N is bounded from below and above by positive constants). Then system (2) is null controllable and the cost of the control C T verifies for T small enough: 1) , for every K > C S (α) := 4

1. If α 2, then C T e K (RT ) 1/(α-1) , for every K > C S (α) := 2 1 α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1 . 2. If α ∈ (1, 2), then C T e K (RT ) 1/(α-
1 α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1
.

(the implicit constant in the previous inequalities might depend on α or R but not on T )

We also have a theorem in the dispersive case when the eigenvalues are not supposed anymore to be positive. Theorem 2.2 Assume that the sequence of increasing eigenvalues (λ n ) n∈Z * of A is a regular sequence of non-zero numbers verifying moreover that there exist some α > 1 and some constant

R > 0 such that        λ n = Rn α + O n→∞ (n α-1 ), n > 0, λ -n = -Rn α + O n→∞ (n α-1 ), n < 0, sgn(λ n ) = sgn(n), (4)
and assume that b n 1. Then system (2) is null controllable and the cost of the control C T verifies for T small enough:

1. If α 2, then C T e K (RT ) 1/(α-1) , for every K > C S+-(α) := 2 α+1 α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1 . 2. If α ∈ (1, 2), then C T e K (RT ) 1/(α-1) , for every K > C S+-(α) := 2 1 α-1 (2 1 α + 1) α α-1 π α α-1 (α -1) 1 α-1 sin( π α ) α α-1
.

(the implicit constant in the previous inequalities might depend on α or R but not on T )

Concerning the parabolic case, we obtain the following result:

Theorem 2.3 Assume that (λ n ) n 1
is a regular increasing sequence of positive numbers verifying moreover that there exists some α > 1 and some constant R > 0 such that (3) holds. Assume that b n 1. Then system (1) is null controllable. Moreover, the control can be chosen in the space C 0 ([0, T ], U ) and the cost of the control C T (in norm L ∞ (0, T ), so this is also true in L 2 (0, T )) verifies for T small enough

C T e K (RT ) 1/(α-1) , for every K > π α α-1 2(α -1) 1 α-1 sin( π 2α ) α α-1
.

(the implicit constant in the previous inequalities might depend on α or R but not on T ) Remark 1

1. Concerning the asymptotic behavior of C S (α), we obtain by easy Taylor expansions that

C S (α) exp(1) 4 (α -1) α+1 1 α-1 as α → 1 + , C S (α) α as α → +∞.
2. We observe that for α = 2, the cost of the control is bounded by e Kπ 2 RT for every K > C S (2) = 2π 2 , which is worse than [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] which stated that we have the same result for every K > 3π 2 /2. However, we see on figure 1 (which compares the upper bound C S (α) and the one found in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] in the case α 2) that our new bound becomes better for α 2.76 and is linearly better as α → ∞. [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] with respect to α.

Another interesting comparison can be done between the upper bound C S (α) and the lower

bound given in [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 transport-diffusion equation[END_REF]. This is what is done in figure 2. We can see that our upper bound becomes close to the lower bound given in [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 transport-diffusion equation[END_REF] when α → ∞, more precisely the difference between the two quantities converges to 1 + ln(2). However, the upper bound is very far from the lower bound for α → 1 + .

Figure 2: Difference between C S (α) and the lower bound of [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 transport-diffusion equation[END_REF] with respect to α.

Remark 2 1. Concerning the asymptotic behavior of C H (α), we obtain by easy Taylor expan-sions that

C H (α) π α α-1 2(α -1) 1 α-1 as α → 1 + , C H (α) α as α → +∞.
2. We observe that for α = 2, the cost of the control is bounded by e Kπ 2 RT for every K > C H (2) = π 2 , which is worse than [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF] which stated that we have the same result for every K > 3π 2 /4. However, we see on figure 3 (which compares the upper bound C H (α) and the one found in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] in the case α 2) that our new bound becomes better for α 2.221 and is linearly better as α → ∞. [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] with respect to α.

3. To finish, let us compare this upper bound and the lower bound given in [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 transport-diffusion equation[END_REF]. This is what is done in figure 4. We can see that our upper bound becomes close to the lower bound given in [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 transport-diffusion equation[END_REF] when α → ∞, more precisely the difference between the two quantities tends to 1. However, the upper bound is very far from the lower bound for α → 1 + .

Figure 4: Difference between C H (α) and the lower bound of [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 transport-diffusion equation[END_REF] with respect to α.

Application to fractional heat or Schrödinger equations with "boundary control"

Let us give the two main applications of this result, which have already been mentioned widely. Let us consider the 1-D Laplace operator ∆ with domain D(∆) := H 1 0 (0, L) and state space H := H -1 (0, L). It is well-known that -∆ : D(∆) → H -1 (0, L) is a positive definite operator with compact resolvent, the k-th eigenvalue is

λ k = k 2 π 2 L 2 , with associated eigenvector e k (x) := sin kπx L /|| sin kπx L || H -1 (0,L) .
Thanks to the continuous functional calculus for positive self-adjoint operators, one can define any positive power of -∆. Let us define our "boundary" control (for more explanations, see notably

[11, Sections 3.3 & 3.4] b ∈ (D(-(-∆) α/2
)) as follows:

b := (∂ x δ 0 ) • ∆ -1 .
We set K := R (for ( 5)) or C (for ( 6)). Let us consider here some γ > 1 and two different equations, one particular case of (1), which are the fractional heat equations

y t = -(-∆) γ/2 y + bu in (0, T ) × (0, L), y(0, •) = y 0 in (0, L), (5)
and one particular case of (2), which are the fractional Schrödinger equations

y t = -i(-∆) γ/2 y + bu in (0, T ) × (0, L), y(0, •) = y 0 in (0, L). (6)
Equation ( 5) is often used to model anomaly fast or slow diffusion (see for example [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF]), whereas [START_REF] Ho | Admissible input elements for systems in Hilbert space and a Carleson measure criterion[END_REF] was introduced to study the energy spectrum of a 1-D fractional oscillator or for some fractional Bohr atoms (see for example [START_REF] Guo | Some physical applications of fractional Schrödinger equation[END_REF]).

As explained in details in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], the above equations exactly fit our abstract setting and we directly deduce from our main Theorems the following results: Theorem 2.4 System (6) is null controllable and the cost of the control C T verifies for T small enough:

1. If γ 2, then C T e KL 2γ/(2γ-1) T 1/(2γ-1) , for every K > 2 1 2γ-1 (2γ -1) 1 2γ-1 sin( π 2γ ) 2γ 2γ-1 . 2. If γ ∈ (1, 2), then C T e KL 2γ/(2γ-1) T 1/(2γ-1) , for every K > 4 1 2γ-1 (2γ -1) 1 2γ-1 sin( π 2γ ) 2γ 2γ-1 .
System (5) is null controllable. Moreover, the control can be chosen in the space C 0 ([0, T ], U ) and the cost of the control C T verifies for T small enough

C T e KL 2γ/(2γ-1) T 1/(2γ-1) , for every K > 1 2(2γ -1) 1 2γ-1 sin( π 4γ ) 2γ 2γ-1 .
(the implicit constant in the previous inequalities might depend on α but not on T ).

Gevrey functions, the moment method and the Bray-Mandelbrojt construction

In this "heuristic" Section, our goal is to give some informal explanations on the strong link between Gevrey functions and the moment method of Fattorini and Russell [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] that will be used here to prove our theorems. Let us consider here the parabolic case (1), the dispersive case being quite similar. Let us decompose our initial condition on our Hilbert basis of eigenfunctions:

y 0 (x) = ∞ k=1 a k e k (x),
with (a k ) k∈N * ∈ l 2 (N * ). Then, using the notations of the previous section, it is well-known that we have for all k ∈ N * and t ∈ (0, T ] that

y(t), e k H = a k e -λ k t + b k t 0 e -λ k (t-s) u(s)ds.
Imposing that y(T, •) = 0 is then equivalent to imposing that for every k ∈ N * , one has

a k e -λ k T + b k T 0 e -λ k (T -s) u(s)dt = 0, i.e. T 0 e -λ k (T -s) u(s)dt = -a k e -λ k T b k ,
the right-hand side being in l 2 (N * ) as soon as b k 1.

Hence, if we assume that we are able to exhibit a bi-orthogonal family to {t → e -λn(T -t) } in L 2 (0, T ), i.e. a family of functions {ψ m } m∈N * such that for every (k, l) ∈ (N * ) 2 one has

e λ k t , ψ l L 2 (0,T ) = δ kl , (7)
then one can use as a control function

u(t) := - k∈N * a k e -λ k T b k ψ k (t). (8)
A usual way (but not the only one, see notably [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]) to construct the family{ψ k } is to use the Paley-Wiener Theorem (see for example [START_REF] Rudin | Real and complex analysis[END_REF]Theorem 19.3, Page 370]), which says that ψ k can be constructed (after some adequate translation) as the inverse Fourier Transform of a L 2 (R)function J k of exponential type T /2 verifying moreover J k (iλ l ) = δ kl . As we will see during the proof, finding such a function J k can be decomposed into two main steps:

• First exhibit some family of Weiertrass product Ψ k involving the eigenvalues and verifying Ψ k (iλ l ) = δ kl (see notably Lemma 3.1).

• Then multiply Ψ k with some adequate function M k (called multiplier ) so that the product J k := Ψ k M k will then be of exponential type T /2 and belongs to L 2 (R). Moreover M k has to verify the normalization condition

M k (iλ k ) = 1. (9)
The crucial point is that the behavior for x → ∞ of the Weierstrass product is very bad and can be proved to be exactly of the form exp(K|x| 1/α ) for some constant K > 0 if we assume that λ k k α for some α > 1. Hence, we observe that if we want to obtain a function J k which is in L 2 (R), it is necessary that the multiplier M k behaves at least as exp(-K|x| 1/α ) at infinity. This exactly means that M k has to be the Fourier transform of some function G k which has to be Gevrey of order α, i.e. verifying

||G (j) k || ∞ C Gev (α, T, R)R j j! α , (10)
for some constant R > 0 > 0 (see for example [19, Section 1.6, Page 30]). Moreover, as it will be observed during the proof of our theorems, an important point is that the constant R determines in an optimal way the exact behavior of the exponential decreasing of M k (i.e. the constant K). Hence, imposing here that M k compensates quite "exactly" the growth of Ψ n means that this constant ν in [START_REF] Lissy | An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit[END_REF] is totally imposed by the data of our problem. To finish, since we want M k to be of exponential type T /2, applying one more time the Paley-Wiener Theorem, it is equivalent to saying that G k has to be of compact support [-T /2, T /2]. Using some estimations made more precise later, the function ψ k will also be bounded by C Gev (α, T, R) and then using [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] we deduce that C H (α, T ) is also bounded by C Gev (α, T, R). Hence, if we want to obtain a precise bound on C H (α, T ), it is necessary to make all our possible to obtain some constant C Gev (α, T, R) which is as small as possible. Let us also mention that the normalization condition ( 9) can be easily replaced by the condition |G k | = 1 (and some suitable inequality on G k (iλ k ) that we do not detail here for the sake of clarity, see notably (60)).

Taking into account all these remarks, the rules of the game can be gathered as the following.

Optimizing the cost of the control is closely related to the following problem: given some T > 0, construct a Gevrey function G of order α, with support equal to [-T /2, T /2], verifying |G| = 1, with imposed coefficient R appearing in the growth of the derivative and minimizing the quantity C Gev (α, T, R) (which has of course to explode as T → 0) that is appearing in [START_REF] Lissy | An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit[END_REF].

Let us now explain a possible way to construct Gevrey function (or more generally C ∞ functions) with compact support. This construction was first described by Szolem Mandelbrojt in [13, Section 13], but in this book it is mentioned that the construction comes from previous unpublished works of Hubert Bray, whence the name "Bray-Mandelbrojt construction" adopted here. The idea is to use repeated mean-values of functions, i.e. to make an infinite convolution product of rectangle functions on [-a k , a k ], (a k ) k 0 being supposed to be a convergent series of positive numbers (whose sum will exactly be the support of the final function). This construction can also be found in [START_REF] Hörmander | The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis[END_REF]. Let us mention that since the Fourier transform of a rectangle function is exactly some cardinal sine function, and that the Fourier transform changes convolution into products, this construction is totally equivalent to the ones we can find notably in [16, Proof of Lemma 4.4], which is also used in [START_REF] Miller | How Violent are Fast Controls for Schrödinger and Plate Vibrations?[END_REF] and that goes back to the work of Ingham in [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] (we refer to [START_REF] Redheffer | Completeness of sets of complex exponentials[END_REF] for extra explanations on the usefulness of this construction).

To finish, let us mention that in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF], the authors used to following Gevrey function to construct the multiplier:

G(x) := exp(-ν/(1 -x 2 )),
where ν > 0 is some large parameter to be chosen. It can be proved that G is a Gevrey function of order exactly 2. Notably, this function is not Gevrey of order α for α ∈ (1, 2) (but it is a Gevrey function of order α for every α > 2), and this explains why in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] (where the same multiplier was used) we were not able to treat the case α ∈ (1, 2) and why our estimations were quite bad for large α.

3 Proofs of Theorems 2.1, 2.2 and 2.3

Estimates on the Weierstrass product involving the eigenvalues

As usual when the moment method is concerned, we first study the asymptotic behavior for large z ∈ C of the Weierstrass product constructed from the family of eigenvalues (λ n ) n∈N . Lemma 3.1 Let (λ n ) n 1 be a regular increasing sequence of positive numbers verifying moreover that there exists some α > 1 and some constant R > 0 such that (3) holds.

Let Φ n be defined as follows:

Φ n (z) := k =n (1 - z λ k -λ n ). (11)
Then,

1. If α ∈ (1, 2), then for every z ∈ C, Φ n (z) e 2 1/α π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), ( 12
)
where P is a polynomial.

2. If α 2, then for every z ∈ C,

Φ n (z) e π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), ( 13
)
where P is a polynomial.

3. If x ∈ R, Φ n (-ix -λ n ) e π 2R 1/(α-1) sin(π/2α) |x| 1 α P (λ n , |x|), ( 14
)
where P is a polynomial.

(In the previous inequalities, the implicit constant may depend on α and R but not on z, x or n)

Remark 3 One can see numerically by taking the particular case λ k = k α that inequalities [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF] and ( 14) are optimal. However, as explained in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]Remark 3] we are unable to extend inequality (13) to the case α ∈ (1, 2), where is seems numerically not to be true anymore. However, it is likely that estimate (12) is far from being optimal (because notably of the gap when α → 2 -with what we have in (13) for α = 2).

Proof of Lemma 3.1. Without loss of generality, we can assume that R = 1 (one can go back to the general case by an easy scaling argument).

Let us prove inequality (12) (inequality ( 13) was already proved in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]). We go back to the computations done in [11, Proof of Lemma 2.1]. One notably has

|Φ n (z)| (1 + |z|/γ) 2C e |z| 0 ∞ γ (λn +s) 1 α -((λn-s) + ) 1 α (t+s) 2 dsdt . ( 15
)
We also have

|z| 0 ∞ γ (λ n + s) 1 α -((λ n -s) + ) 1 α (t + s) 2 dsdt = |z| ∞ γ (λ n + s) 1 α -((λ n -s) + ) 1 α s(s + |z|) ds |z| λ 1-1 α n (U ( |z| λ n ) + V ( |z| λ n )), (16)
where

U (x) := 1 0 (1 + v) 1 α -(1 -v) 1 α v(v + x) dv and V (x) := ∞ 1 (v + 1) 1 α v(v + x) dv.
Using the change of variables t = x/v, we easily obtain

x 1-1/α V (x) = x 0 (1/t + 1/x) 1 α 1 + t dt (17)
and

x 1-1/α U (x) = ∞ x (1/t + 1/x) 1 α -(1/x -1/t) 1 α 1 + t dt. ( 18
)
Using expression [START_REF] Miller | On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian[END_REF], one has

x 1-1/α V (x) 2 1 α x 0 1 t 1 α (1 + t) dt.
Using expression [START_REF] Redheffer | Completeness of sets of complex exponentials[END_REF], we also have

x 1-1/α U (x) 2 1 α ∞ x 1 t 1 α (1 + t) dt.
Hence we deduce [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 transport-diffusion equation[END_REF] taking into account that (see for example [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]

, Lemma 2.2]) ∞ 0 dt t 1 α (1 + t) = sin( π α ). (19)
Concerning ( 14), one verifies that the proof provided in [11, Page 2661] for α 2 is also valid for α ∈ (1, 2) because we can use [START_REF] Robino | Linear partial differential operators in Gevrey spaces[END_REF], replacing α by 2α (> 1).

In the dispersive case, we also need some estimate in the case where the eigenvalues are not supposed to be positive anymore. Lemma 3.2 Let (λ n ) n 1 be a regular increasing sequence of positive numbers verifying moreover that there exists some α > 1 and some constant R > 0 such that (4) holds.

Let Φ n be defined as follows:

Φ n (z) := k =n (1 - z λ k -λ n ). (20)
Then,

1. If α ∈ (1, 2), then for every z ∈ C, Φ n (-z -λ n ) e (2 1/α +1)π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), (21)
where P is a polynomial.

2. If α 2, then for every z ∈ C,

Φ n (-z -λ n ) e 2π R 1/(α-1) sin(π/α) |z| 1 α P (λ n , |z|), (22)
(In the previous inequalities, the implicit constant may depend on α and R but not on z, x or n)

This Proposition was already proved in [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF] for the case α 2. For α ∈ (1, 2), the proof is exactly the same as the corresponding case in Lemma 3.1 that we combine with the computations made in the proof of [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]Lemma 2.4 ] (see Pages 2666-2668 in this reference) and will be omitted.

Construction of adequate multipliers

As explained before, we now have to construct an adequate multiplier, which has to be the Fourier transform of a Gevrey function with compact support. Let us emphasize that the main contribution of this paper is the construction given in (43).

To begin, let us give some useful estimates concerning the Bray-Mendelbrojt construction of functions with compact support (see [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF]Section 13]). Proposition 3.1 Let (a k ) k 0 be a decreasing sequence of positive numbers such that

a := k a k < ∞.
Then, there exists a nonnegative function u with compact support included in [-a, a] verifying

a -a u = 1, (23)
u(0) 1 2a 0 , (24)
u is even,

u is increasing on [-a, 0], (25)
such that for every j ∈ N, one has

||u (j) || ∞ 1 j i=0 a i , (27)
Proof of Proposition 3.1 We follow step by step the proof given in [START_REF] Hörmander | The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis[END_REF]. For every b > 0 we call

H b := 1 [-b,b] 2b . ( 28
) Let us remark that R H b = b -b H b = 1 (29)
and H b is even.

We then consider

u n := H a0 * H a1 * • • • * H an ,
where * represents the convolution product. u n is of class C n-1 and

Supp(u n )[- n 1 a i , n 1 a i ](⊂ [-a, a]). (31)
Moreover, one easily verifies that, for j n -1 we have

u (j) n (x) = j-1 0 1 2a i (τ -ai -τ ai ) H aj * • • • * H an (x), (32)
where

τ b : u → (x → u(x -b)).
Using (29), the fact that for u and v some regular enough functions one has u * v = u v, and |u * v| ||u|| ∞ ||v|| 1 , we deduce, taking into account (32), that

||u (j) n || ∞ 1 j i=0 a i .
Let us also remark that u n is even (33) and verifies

a -a u n (x)dx = H a1 • • • H an = 1, (34)
moreover

u n (0) = a -a H a0 (s)H a1 * • • • * H an (-s)ds = 1 2a 0 a0 -a0 H a1 * • • • * H an (-s)ds 1 2a 0 a -a H a1 • • • a -a H an = 1 2a 0 . (35)
Let us now prove that u n is increasing on [-a, 0] for every n ∈ N. Let us prove this by induction. it is true for n = 0 and n = 1. Let us assume that u n is increasing on [-a, 0]. Then we compute the derivative of u n+1 , we obtain

u n+1 (x) = 1 2a n+1 (τ -an+1 -τ an+1 )(H a0 * • • • * H an (x)) = 1 2a n+1 (u n (x+a n+1 )-u n (x-a n+1 )). ( 36
)
Let x ∈ [-a, 0]. Then, we distinguish two cases:

1. If x -a n+1 , then x -a n+1 x + a n+1 0, hence, by using the fact that u n is increasing on [-a, 0], it is clear that u n (x -a n+1 ) u n (x + a n+1 ), and then by (36), u n+1 (x) 0.

If

x -a n+1 , then we have -a n+1 -x 0. But we also have a n+1 -x -x -a n+1 0.

Using the fact that u n is even, we know by induction that u n is decreasing on [0, a], hence we deduce that

u n (x + a n+1 ) = u n (-x -a n+1 ) u n (a n+1 -x) = u n (x -a n+1 ),
and then by (36), u n+1 (x) 0.

We conclude as in [START_REF] Hörmander | The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis[END_REF] by letting n → ∞, u being the limit (that exists) of the u n .

According to Section 2.3 and to inequality (27) (that has to be verified for every n), we see that the problem of minimizing what we called C Gev (α, T, R) in the case of the Bray-Mandelbrojt construction can be reformulated as maximizing the following quantity with respect to the sequence (a n ) n∈N :

max j∈N j i=0 (i + 1) α a i ν α-1 ,
for some large parameter ν > 0 to be chosen later. It is clear that this problem is equivalent to maximizing

∞ i=0 (i + 1) α a i ν α-1 .
Taking into account all these considerations, it is quite natural to investigate the following optimization Problem.

Problem (1) Let ν > 0 and a > 0. Find the sequence (a k ) k∈N that maximizes the quantity

∞ k=0 ln (k + 1) α a k ν α-1
with the constraints:

1. a k > 0 for every k ∈ N.

2. (a k ) k∈N is non-increasing.

3.

∞ k=0 a k = a. (37)
Unfortunately, we have the following proposition: Proposition 3.2 There are no solutions to Problem [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].

Proof of Proposition 3.2. Let f ((a k )) := ∞ k=0 ln (k + 1) α a k v α-1 . (38)
One has

∂f ∂a i ((a k )) = 1 a k . ( 39
) Let g((a k )) := ∞ k=0 a k . (40)
One has

∂g ∂a i ((a k )) = 1. (41)
We want to maximize f under the constraint g((a k )) for every j ∈ N (the constraint a k > 0 can be forgotten here). We do not need here to use that we want (a k ) k∈N to be decreasing.

Let λ ∈ R and (µ j ) j∈N , we consider the Lagrangian

L((a k )) := f ((a k )) -λg((a k )). (42)
We remark that for every j ∈ N, we have ∂L ∂aj = 0 if and only if a j = λ according to (39) and (41), hence the sequence (a j ) has to be constant. If we take into account the fact that we want the constraint g((a k )) = a to be verified, we see that Problem (1) cannot have any solution.

We observe thanks to the previous study that if we want to "mimic" in some sense an optimal sequence, a good idea would be to consider some sequence (a k ) k∈N which is constant at least for small k and then decreases in some suitable way such that it becomes summable. Hence, taking into account this remark, we set ν > 0 some parameter (that is destined to compensate the bad growth of the Weierstrass product [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], see (59), and to be very large so we will always assume ν C > 0), and we consider the following sequence (a k ) k∈N defined by

a k = 1 ν if k ν -1, a k = ν α-1 (1+k) α if k ν , (43)
The main advantage of this construction is that for k ν , we have

(k + 1) α a k v α-1 = 1,
which will simplify a lot the forthcoming computations.

Remark 4

The second part of the construction of (43) may seem to be chosen quite arbitrarily, however the author did not find a more suitable construction. It might be possible that another appropriate decreasing that would enable to improve the bounds given in this article.

We consider the corresponding function σ ν constructed as in the proof of Lemma 3.1 from the sequence (a k ) k∈N . We call a ν its support, that is to say

a ν := 1 + ∞ k= ν ν α-1 (1 + k) α . (44)
It may seem quite strange that we do not impose the support to be [-T /2, T /2] here, however for the sake of clarity we prefer to adjust the support thanks to the parameter β to be introduced later (see (58)).

Taking into account the definition of a ν , we may observe that a ν does not depend too much on a ν in the sense that

a ν ∼ α α -1 as ν → ∞. (45)
Let us remark that

aν aν σ ν = 1, (46)
σ ν (0) ν 2 , ( 47
)
σ ν is even, ( 48
)
σ ν is increasing on [-a v , 0]. (49)
Then, using (27), we obtain

||u (j) || ∞ 1 j i=0 a i . (50)
We can then deduce the following crucial estimate :

Lemma 3.3 One has for every j ∈ N ||u (j) || ∞ e αν (j!) α (ν α-1 ) j . (51)
Now we choose ν such that α α (βν α-1 ) > π sin( π α ) and close to π/ sin(π/α) (see estimates [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF] and (59)), for example ν := (π + δ) α/(α-1) (α sin(π/α)) α/(α-1) β 1/(α-1) .

(69)

Then, using (67), ( 13), ( 68), ( 69) and (59), we obtain

|g n (x)| e αν+π/ sin(π/α)|x+λn| 1 α -(π+δ/2)/ sin(π/α)|x+λn| 1 α P (|x + λ n |).
It is clear that

e π/ sin(π/α)|x+λn| 1 α -(π+δ/2)/ sin(π/α)|x+λn| 1 α P (|x + λ n |) 1 1 + (x + λ n ) 2 ,
hence we deduce using (69) and (68) that

|g n (x)| e αν 1 + (x + λ n ) 2 e α(2α) 1/(α-1) (π+δ) α/(α-1) (α sin(π/α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1) 1 + (x + λ n ) 2 e 2 1/(α-1) (π+δ) α/(α-1) (sin(π/α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1) 1 + (x + λ n ) 2 .
Let us fix some K > 2 1/(α-1) π α/(α-1) (α -1) 1/(α-1) sin(π/α) α/(α-1) T 1/(α-1) . Considering δ as close as 0 as needed, we deduce that

|g n (x)| e K T 1/(α-1) 1 + (x + λ n ) 2 (70)
This notably proves that g n ∈ L 2 (R). Moreover, using ( 13), ( 67 The case α ∈ (1, 2) follows by doing exactly the same proof, the only difference being that we use [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 transport-diffusion equation[END_REF] instead of [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF].

Proof of Theorem 2.2. The proof is exactly the same as the one of Theorem 2.1, using (21) instead of [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 transport-diffusion equation[END_REF] for α ∈ (1, 2) and ( 22) instead of (13) for α 2, and will hence be omitted.

Proof of Theorem 2.3. We now consider the parabolic case (Equation ( 1)). Let δ > 0 a small enough parameter. We call

h n (z) := Φ n (-iz -λ n ) H β (z) H β (iλ n ) . ( 72
)
One has, using [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], that h n (iλ k ) = δ kn . We want to apply at the end the Paley-Wiener Theorem (see estimate (58)) in an optimal way, so we want a ν β to be close to T /2. Let us consider ν large enough such that (see ( 45))

|a ν - α α -1 | δ/4.
Assume that a ν β < T /2 and close to T /2, for example

β = T (α -1)(1 -δ/2) 2α . ( 73
)
Now we choose ν such that α α (βν α-1 ) > π/(2 sin(π/2α))

and close to π/(2 sin(π/(2α))) (see estimates [START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF] and (59)), for example ν := (π + δ) α/(α-1) (2α sin(π/2α)) α/(α-1) β 1/(α-1) .

(74)

Moreover, thanks to (72), ( 14), (59), (60), ( 74) and (73), one has |h n (x)| νe αν+π/(2 sin(π/2α))|x| (2 sin(π/2α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1)

(1 + (x + λ n ) 2 ) .
Let us fix some K > π α/(α-1) 2(sin(π/2α)) α/(α-1) (α -1) 1/(α-1) .

Considering δ as close as 0 as needed, we deduce that

|h n (x)| e K T 1/(α-1) (1 + (x + λ n ) 2 ) , (75)
This notably implies that h n (x) ∈ L 1 (R) ∩ L 2 (R) and

Figure 1 :

 1 Figure 1: Difference between C S (α) and the upper bound of [11] with respect to α.

Figure 3 :

 3 Figure 3: Difference between C H (α) and the upper bound of [11] with respect to α.

  ), (68) and (58), we obtain|g n (z)| e T |z|/2 .Hence, using the Paley-Wiener Theorem, g n is the Fourier transform of a function f n ∈ L 2 (R) with compact support [-T /2, T /2]. Moreover, by construction {f n } is biorthogonal to the family {e iλnt }. Then, one can create the control thanks to the family {f n }. Let us consider y 0 = a k e k the initial condition, we callu(t) := -k∈N (a k /b k )e -iT λ k /2 f k (t -T /2). (71)This expression is meaningful since b k 1, moreover the corresponding solution y of (2) verifies y(T, •) ≡ 0. We obtain by proceeding as in [22, Page 87-88] that ||u(t)|| L 2 (0,T ) α-1) ||y 0 || H .

1 α

 1 -((π+δ/2)/(2 sin(π/2α)))|x| 1 α -βλn 4ν P (|x|, |λ n |) νe αν-δ/(2 sin(π/2α))|x| 1 α -βλn 4ν P (|x|, λ n |) ν e α (π+δ) α/(α-1)(2α sin(π/2α)) α/(α-1) β 1/(α-1) (1 + (x + λ n ) 2 ) ν e α (2α) 1/(α-1) (π+δ) α/(α-1) (2α sin(π/2α)) α/(α-1) (T (α-1)(1-δ/2)) 1/(α-1) (1 + (x + λ n ) 2 )ν e 2 1/(α-1) (π+δ) α/(α-1)

  ||h n || L 1 (R) e K T 1/(α-1) .(76)Moreover, using[START_REF] Mandelbrojt | Analytic functions and classes of infinitely differentiable functions[END_REF], (72), (58) and (73)|h n (z)| e T |z|/2 , so using the Paley-Wiener Theorem, h n is the Fourier transform of a function w n ∈ L 2 (R) with compact support [-T /2, T /2]. Moreover, by construction {w n } is biorthogonal to the family {e λnt }. Then, one can create the control thanks to the family {h n }. Let us consider y 0 = a k e k the initial condition, we callu(t) := -(a k /b k )e -T λ k /2 w k (t -T /2),(77)This expression is meaningful since b k 1, moreover the corresponding solution y of (1) verifies y(T, •) ≡ 0. One easily verifies that u ∈ C 0 ([0, T ], R). Using (77), |b k | 1 and inequality (76), we obtain ||u(t)|| L ∞ (0,T ) e K T 1/(α-1) |a k |e -T λ k /2 . Using the Cauchy-Schwarz inequality, one deduces ||u(t)|| L ∞ (0,T ) e K T 1/(α-1) ||y 0 || H .

Proof of Lemma 3.3. Using the expression of a k given in (43) and (50), we deduce that for j ν we have

ν α-1 /(i + 1) α (i + 1) α a i /ν α-1 (53)

Now, using an usual lower bound on the factorial, we obtain

from which we deduce (51).

The same estimation is also clearly true for j ν since in this case we have

which will be our multiplier (up to some homothety). We have the following properties:

Lemma 3.4 For every δ > 0 small enough, one has

Proof of Lemma 3.4. Equality (57) comes from (46). Inequality ( 58) is an immediate consequence of the definition of H β given in (56). Let us now prove (59). Since all derivatives of σ ν vanish at t = -a ν and t = a ν , we have

for every x > 0 and j ∈ N. Combining (61) and (51), we deduce that

We set

Then we have

Using ( 64) and ( 62) we obtain

Combining ( 65), (63), and inequality

we deduce |H β (x)| e αν e -αj j α/2 e αν e -α(βν α-1 x) 1/α +δ/(2 sin(π/(α)))x 1/α , which proves the desired estimate.

Let us know prove (60). We have, since σ ν 0 and using (49) together with (48),

Using ( 46), ( 47), ( 48) and ( 49), we have

We deduce that for ν large enough (which is equivalent to T small enough, see (69))

which gives the desired result thanks to (66).

Proof of Theorem 2.1. The proof follows the one of [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]Theorem 3.1 and 3.4]. We still assume without loss of generality that R = 1. Let us first consider the dispersive case (Equation ( 2)) and the case α 2. Let δ > 0 a small enough parameter. We call

so that one has g n (-λ k ) = δ kn by ( 57) and [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF]. We want to apply at the end the Paley-Wiener Theorem (see estimate (58)) in an optimal way, so we want a ν β to be close to T /2. From now on we will always consider ν large enough such that (see (45))

Assume that a ν β < T /2 and close to T /2, for example